AN 
ELEMENTARY TREATISE 
ON 
OPTICS, 
DESIGNED FOR THE USE OF THE CADETS 
OF THB. 
' ; ■ '* '• *, (• 
UNITED STATES’ MILITARY ACADEMY. 
BY WM. H. C. BARTLETT, A. M., 
PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY 
IN THE ACADEMY. 
NEW-YORK: 
WILEY AND PUTNAM, 161 BROADWAY. 
1839 Entered according to the act of Congress, in the year 1839, 
* BY WILLIAM H. C. BARTLETT, A. M., 
Jn the Clerk’s office of the District Court of the Southern District of New-York. 
Printed by William Osborn,* 
88 William-street. PREFACE, 
In presenting the following pages to the public, 
their author advances no claims to originality. His 
only object has been to prepare, what appeared to 
him, a suitable elementary class book on the sub- 
ject of which they profess to treat, for the use of 
the Cadets of the United States’ Military Academy. 
In doing this, he has availed himself, principally, 
of the works of Mr. Coddington, Sir David Brew- 
ster, Sir John Herschell, and the Rev. Baden 
Powell; and to these distinguished authors, he 
would acknowledge his obligation for whatever 
of merit this little volume may be found to possess. CONTENTS. 
Pa g* 
Introductory remarks and definitions, . 1 
Reflection and refraction, . . . 1 . 4 
Fundamental laws, ..... 5 
Table of refractive indices and refractive powers, . 8 
Deviation at plane surfaces, ... 9 
Deviation at spherical surfaces, . . .20 
Deviation of small direct pencil at spherical surfaces, 22 
Spherical lens, . . . . . .26 
Power of a lens, ..... 30 
Deviation by refraction through the various kinds of 
spherical lenses, . . . . .32 
Deviation by reflection at spherical reflectors, . . 39 
Spherical Aberration, .... 43 
Deviation of a small oblique pencil, . . .57 
Oblique pencil through the optical centre, . . 58 
Oblique pencil not through the optical centre, . . 59 
Optical images, ..... 66 
Caustics, . . . • • .76 
Surfaces of accurate convergence, . . . 87 
The eye and vision, . . - . .93 
Apparent magnitudes of objects, . . .99 
Microscopes and telescopes, . . . 101 
Common astronomical telescope, . . 108 
Galilean telescope, .... 109. 113 
Terrestrial telescope, . . . .114 
'Compound refracting microscope, . 114 vi 
CONTENTS. 
Page 
Reflecting telescope, . . . .116 
Herschelian telescope, . . . . 117 
Gregorian telescope, . . . 117 
Cassegrainian telescope, - - . 119 
Newtonian telescope, . . . . 119 
Dynameter, . . ' . . 119 
Micrometer, ...... 123 
Sextant, ..... L27 
Adjustments of sextant, . . . . 129 
Artificial horizon, .... 133 
Camera lucida, . . . . .134 
Camera obscura, ..... 135 
Magic lantern, ..... 137 
Solar microscope, .... 138 
Unequal refrangibility of light, . . .138 
Dispersion of light, .... 142 
Table of dispersive powers, . . . 145 
Chromatic aberration, .... 146 
Achromatism, . . . . .149 
Absorption of light, . . . . 155 
Internal reflection, ..... 161 
Rainbow, . . . . . 163 
Halos, ...... 173 
Interference of light, . . . . 174 
Divergence of light, * .... 185 
Colored fringes of shadows and apertures, . 186 
Double refraction, ..... 190 
Polarization of light, .... 197 
Polarization by reflection, .... 198 
Polarization by refraction, . . . 205 
Polarization by absorption, .... 206 
Polarization by double refraction, . . 207 
Interference of polarized light, . . . 209 
Circular polarization, . 222 
Elliptical polarization, .... 224 
Colors of thin plates, . . . 225 ELEMENTARY TREATISE 
ON 
OPTICS. 
1. Light is that principle by whose agency we 
derive our sensations of external objects through 
the sense of sight. 
2. That branch of Natural Philosophy which 
treats of the nature and properties of light, is 
called Optics. 
3. All bodies are divided with respect to light, 
into two classes, called Self-luminous and Non- 
luminous. 
4. Self-luminous bodies are such as possess the 
power of exciting light; as the sun, stars, &c. 
Generally, all substances become self-luminous 
when their temperature is sufficiently raised. 
5. Noil-luminous bodies are such as do not possess 
the power of exciting light, and are visible in con- 
sequence only of light derived from bodies of the 
self-luminous class. 
1 2 
PRELIMINARIES. 
6. Astronomical observations have shown that 
the communication which light produces between 
us and luminous objects, is not instantaneous. 
When the sun, for example, is in any assumed 
point of his orbit, the sensation of his presence 
there is not communicated to us till 8' 13" after- 
wards. Knowing the distance of the earth from 
the sun, the velocity with which light moves will 
easily result. It is found to be about 195,200 
miles a second. 
7. This extremely rapid communication is sup- 
posed to be made either by pulsations transmitted 
through a highly elastic fluid, as sound is transmit- 
ted through the air; or by real emanations of 
material particles thrown oft' from the surfaces of 
luminous bodies. Since we see objects through a 
certain class of bodies called transparent, it would 
follow that the pulsations or waves of the elastic 
fluid continue to be propagated through the inter- 
stices of these substances, or that the luminous 
particles, on the supposition of material emanations, 
pass through the same openings. 
The eye admitting the free passage of light 
into it, the sensation of vision is supposed to arise, 
on the supposition of material emanations, from 
the action of the particles on certain nerves which 
are spread over the inner surface of the back 
part of that organ. In the theory of elastic fluid, 
vision is attributed to vibratory motion communi- 
cated to the same nerves by the pulsations or PRELIMINARIES. 
3 
waves propagated through the elastic fluid, with 
which they are in contact. 
To give any thing like an adecjuate idea of the 
theories here referred to, would far transcend the 
limits of an elementary treatise; nor is it neces- 
sary to our present purpose, to know any thing with 
respect to the real nature of light. The investiga- 
tions pursued in the following pages being founded 
upon data derived from actual experiment, the re- 
sults will be true whatever changes may take place 
in the theories with respect to the nature of the 
agent or principle to which these results refer. 
8. A ray of light, denotes any rectilinear direc- 
tion in which the effect of light is conveyed. It 
will be convenient, hereafter, to associate with a 
ray, the idea of motion, and wherever this is done, 
nothing more is meant than a reference to the suc- 
cessive occurrences of the effect of light at the dif- 
ferent points along the ray. 
9. A collection of parallel rays, is called a beam 
of light. 
10. A collection of rays diverging from, or con- 
verging to a point, is called a pencil of light. 
11. Whatever affords a passage to light is called 
a medium, such as glass, water, air, vacuum, &c. 
12. Light is transmitted from one point to an- 
other, in the same medium of homogeneous den- 4 
REFLECTION AND 
sity, in right lines; for if an opaque body be inter- 
posed between the eye and an object to which it is 
directed, the object will be concealed from view. 
Reflection and Refraction of Light. 
13. When a beam or pencil of light S D, is inci- 
dent upon any surface M N, (fig. 1), separating two 
media of different density, such as air and water, 
for instance, universal experience has shown that 
the beam or pencil will be divided into two por- 
tions, one of which will be driven back from the 
surface M N, in some direction as D S', and con- 
tinue in the same medium, while the other will 
penetrate the surface and be transmitted through 
the second medium in some direction as D S". 
The first is said to be reflected, and the second 
refracted. 
The circumstances attending these two portions 
of light being in general different, gave rise to two 
distinct branches of optics, viz., Catoptrics, and 
Dioptrics; the former treating of reflected, and 
the latter of refracted light. But these will be 
considered in connection, as by that means much 
time and space will be saved, and the discussion 
rendered general. 
14. The line P D P', (fig. 2), being supposed 
normal to the surface of which M N is a section 
by a normal plane, and the light to proceed in the 
direction from S to D, S D is called the incident 
ray, D S' the reflected ray, and D S" the refrac- REFRACTION OF LIGHT. 
5 
ted ray. The angle S D P is called the angle of 
incidence, P D S' the angle of reflection, and P' 
D S" the angle of refraction. 
15. All change in the direction of the incident 
light, either by reflection or refraction, is found to 
take place immediately at the surface separating 
the two media. If the surface of separation be 
curved, as M'N' or IVT'N", we may conceive a tan- 
gent plane to be drawn through the point of inci- 
dence D, when the beam of light being regarded 
indefinitely small, the angles of incidence, reflec- 
tion and refraction will remain unchanged. 
16. Experiments have shown, 
First, That the incident ray, reflected ray, and 
refracted ray are always, except in a particular 
case to be noticed hereafter, contained in the same 
plane normal to the surface separating the media. 
Second, That the sine of the angle of incidence 
is equal to that of reflection. 
Third, That, for the same medium, the sine of 
the angle of incidence hears to the sine of the angle 
of refraction, a constant ratio. 
If to these facts, which are the results of careful 
experiment, we add the rectilineal propagation of 
light, we shall have all the fundamental laws upon 
which the whole mathematical theory of Catoptrics 
and Dioptrics depends. 
17. Denoting by v the angle of incidence; by T 
that of refraction; and by m the constant ratio of 6 
REFLECTION AND 
the sines of these angles, the third law will be ex- 
pressed by the equation, 
sin q> = m sin cp' . . . (1) 
The angle which any ray makes with the nor- 
mal, will be estimated from that part of the normal 
lying in the medium with the ray, and in a direc- 
tion towards the incident ray, from the part of the 
normal nearest to it. Thus, (fig. 3), the angle 
which the reflected ray D S' makes with the nor- 
mal, is equal to 360°—P D S' ; while the angle 
made by the refracted ray D S", is P'D S", being 
estimated from D P', in the same direction around 
D. By this convention, we shall be able to con- 
vert all expressions relating to refraction into others 
appertaining to reflection, by simply changing m 
into — 1. This in equation (1), gives 
sin (p = — sin q>' . . (2) 
which expresses the second law; or, to include both 
reflection and refraction under the same formula, 
sin q> ~ ± m sin q>'. 
18. M'N' (fig. 4), being a section of the sepa- 
rating surface, M N that of a tangent plane at 
the point of incidence D, P D P' the normal, S D 
the incident, D S' the reflected, and S D" the 
refracted rays ; the angle S' D Q, made by the 
reflected ray with its direction before incidence, 
is called the deviation by reflection, and S"DQ, 
the deviation by refraction; the rays are said to be REFRACTION OF LIGHT. 
7 
deviated at the point D; and the surface of which 
M' N' is a section, is called the deviating surface. 
19. The numerical value of m which expresses 
the quotient arising from dividing the sine of inci- 
dence by the sine of refraction, although constant 
for the same medium, varies from one medium to 
another. As a general rule, it is greater than unity 
when light passes from any medium to another of 
greater density, such as from air to water, from 
water to glass, &c. ; and less than unity when 
light passes from any medium to one less dense, 
as from water into air. 
There is a remarkable exception to this rule in the 
case of combustible substances, these always refrac- 
ting more than other substances of the same density. 
From what has been said, it is obvious that a ray 
of light on leaving any medium and entering one 
more dense, will be bent towards the normal to 
the deviating surface, while the reverse will be the 
case when the medium into which the ray passes is 
less dense than the other. 
The numerical value of m, has been determined 
for a great variety of substances, solids liquids and 
gases, on the supposition of the deviating surface 
being that which separates the various substances 
considered from a vacuum. If all bodies possessed 
equal density, the value of m, or the index of re- 
fraction, might be taken as the measure of the re- 
fractive power of the substance to which it belongs, 8 
REFLECTION AND 
but this not being the case, Sir Isaac Newton has 
shown, that on the supposition of the law accord- 
ing to which all substances act upon light being of 
the same form, the refractive power will be propor- 
tional to the excess of the square of the index above 
unity, divided by the specific gravity. Calling n 
the absolute refractive power, m, the index of re- 
fraction, S, the specific gravity, and A, a constant co- 
efficient, we shall have according to this rule, 
m2—1 
n—A. g ... (3) 
The following table shows the value of m, and 
n, for the different substances named, the value of m 
being taken on the passage of light from a vacuum. 
Table of Refractive indices and Refractive Powers, 
Substances. 
m 
m2—1 
*=-s 
Chromate of Lead, 
( 2.97 
\ 2.50 
1.0430 
Realgar, 
2.55 
1.066 
Diamond, 
2.45 
1.4566 
Glass-flint, 
1.57 
0.7986 
Glass Crown, 
1.52 
Oil of Cassia, 
1 (33 
1.3308 
Oil of Olives, 
1.47 
1.2607 
Quartz, 
1.54 
0.5415 
Muriatic Acid, 
1.40 
Water, 
1.33 
0.7845 
Ice, 
1.30 
Hydrogen, 
1.000138 
3.0953 
Oxygen, 
1.000272 
0.3799 
Atmospheric Air, 
1.000294 
0.4528 ON THE DEVIATION OF LIGHT, &c. 
9 
On the Deviation of Light at Plane Surfaces. 
20. Let M N, (fig. 5), be a deviating surface, 
separating any medium B from a vacuum A. 
A ray of light S D, being incident at D, will be 
deviated according to the law expressed by equa- 
tion (1). 
sin cp =*m sin cp1 
m being the index of refraction of the medium B. 
The refracted ray D D', meeting a second surface 
M' N', parallel to the first, and passing again into 
a vacuum, will be refracted so as to satisfy the 
equation, 
sin cp' — m1 sin cp" 
the angle of incidence on the second surface being 
the same as that of refraction at the first, and de- 
noting by m! the index of refraction from the me- 
dium B to the vacuum. But, in this case, expe- 
riment gives, 
1 
m = — 
m 
whence we obtain, by means of the foregoing 
equations, 
sin cp — sin cp" 10 
ON THE DEVIATION OF LIGHT 
that is, the ray after passing a medium bounded 
by parallel plane faces, is not deviated, but remains 
parallel to its first direction. 
The ray D" D'", being supposed to traverse a 
second medium bounded by plane parallel faces, 
and of which the refractive index is m", will under- 
go no deviation ; and the same may be said of any 
number of media bounded by similar faces. If, 
now, the spaces between the media be diminished 
indefinitely so as to bring them into actual contact, 
experiment shows there will still be no deviation, 
which might have been inferred. 
21. Let us next suppose, (fig. 6), a ray to tra- 
verse two media, bounded by plane parallel faces, 
the media being in contact, and having their refrac- 
tive indices denoted by m and m! ; we shall have 
by calling m", the index of refraction of the second, 
or denser medium in reference to the first, 
sin cp — m sin cp' 
sin cp' = m" sin cp1' . . . (4) 
„ 1 . 
sin op = —7 sin op. 
m 
Multiplying these equations together, there will 
result 
„ m 
m = —, 
m 
That is, to find the index of refraction where a ray AT PLANE SURFACES. 
11 
passes from any one medium to another, divide 
the index of the second by that of the first, referred 
to a vacuum. 
22. If a ray pass from a medium to another 
more dense (fig. 7), the index m" will be greater 
than unity, and from equation (4), we shall have 
sin cp' > sin 9", 
and if sin <p' be taken a maximum, or the angle of 
incidence be 90°, equation (4) will give, 
1 
— = sin cp" . . . (5) 
m 
from which results a maximum limit of the angle of 
refraction. If m" be taken equal to 1*52 for the at- 
mosphere and crown glass, 
sin g>" = ‘657, 
or 
cp" =41° 5' 30 nearly ; 
for air and water, m" = 1*33, and 
<p" =48°. 15; 
that is to say, the greatest angle of refraction that 
can exist when light passes from air into crown 
glass, is 41°. 5'. 30" ; and from air into water, 
48° 15'. 12 
ON THE DEVIATION OF LIGHT 
If the ray pass from a medium to another less 
dense, (fig. 8), m" will be less than unity, and 
equal to the reciprocal of its former value ; equa- 
tion (4) will then give, 
sin cp" y sin <j>' 
taking the maximum value for sin v'' — 1, we shall 
obtain from same equation, 
sin cp't— — 
1 m 
this value of the sine of the angle of incidence, 
which corresponds to the greatest angle of refrac- 
tion, when light passes from any medium to one 
less dense, is the same as that found before for the 
greatest angle of refraction when the incidence 
was taken a maximum in the passage of light from 
one medium to another of greater density. 
This value in the case of air and glass, is *657; 
corresponding to an angle of 41°. 5'. 30" ; and for 
air and water, the angle is 48°. 15'. 
If the angle v1 be taken greater than that whose 
l 
sine is the angle of refraction, or emergence 
from the denser medium, will be imaginary, and 
the light will be wholly reflected at the deviating 
surface. This maximum value for <p', is called 
the angle of total reflection. Light cannot, there- 
fore, pass out of crown glass into air under a 
greater angle of incidence than 41°. 5'. 30" ; nor AT PLANE SURFACES. 
13 
out of water into air under a greater angle than 
48°. 15'. 
The maximum limit of refraction, and the case 
of total reflection, are attended with many inte- 
resting results. If an eye be placed in a more re- 
fracting medium than the atmosphere, as that of a 
fish under water, it will perceive, by the limit of 
refraction, all objects in the horizon elevated in the 
air, and brought within 48°. 15' of the zenith, while 
objects in the water would appear to occupy the 
belt included between this limit and the horizon by 
total reflection. 
Those remarkable cases of mirage, where ob- 
jects are seen suspended in the air, and oftentimes 
inverted, are explained by ordinary refraction and 
total reflection. It is well known that in the ordi- 
nary state of the atmosphere, its density decreases 
as we ascend ; a ray of light, therefore, entering 
near the horizon, would undergo a series of refrac- 
tions, and reach the eye with an increased inclina- 
tion to the surface of the earth ; or would appear to 
come from a point in the heavens above that occu- 
pied by a body from which it proceeded. Hence, 
the effect of the atmosphere is to increase the ap- 
parent altitude of all the heavenly bodies. 
Dr. Wollaston suggested a method, founded on 
the limit of total reflection, to determine the re- 
fractive powers of different substances. If the 
angle of incidence , be measured by any device, 
equation (1) will give, 
1 
m = —: — 
sm cp 14 
ON THE DEVIATION OF LIGHT 
from which the absolute refractive power may be 
deduced by equation (3). 
23. The deviating surfaces have, thus far, been 
supposed parallel. If they be inclined to each 
other, as M N, MN', (fig. 9), we shall have what 
is called an oqjtical prism, which consists of any 
refracting substance bounded by plane surfaces in- 
tersecting each other. 
M N and M N' are called the deviating planes, 
and the angle under which they are inclined, is 
called the refracting angle of the prism. 
24. Required the deviation of a ray of light on 
passing through a prism. 
Let SD (fig. 10), be the incident, D D' the first, 
and D' S' the second refracted rays. The total 
deviation will be S E S", which will be called 8; 
calling the refracting angle of the prism «, and 
adopting the notation of the figure, we shall have 
5=kdd'-(-ed'd = 9 — <p' -{-V — = <r+ V7 — W + V'*) 
71 J[ 
180° or ?r = a -|- M D D -f md1 d = a -f- g — <J>1 + g— V7’ 
or 
«= «/'' + <?' • • ■ • (6) 
hence 
d = cp -f- xjj — a . . . (7) 
The deviation of a ray of light on passing 
through a prism is, therefore, equal to the sum of AT PLANE SURFACES. 
15 
the angles of incidence and emergence, diminished 
by the refracting angle of the prism. 
The refracting angle for the same prism being 
constant, the deviation will depend upon the angles 
of incidence and emergence. 
25. Required the relation between the angles of 
incidence and emergence in order that the devia- 
tion shall be a minimum. 
For this purpose we have equations 
fi ~ cp —{— ip— « . . . . (7) 
« = + + 9 (6) 
sin 9 ■= m sin qp' (1) 
sin 9 = vt sin y/ (1) 
From equation (7), we have 
d . S = dcp -{- dtp = + 1) d (p = 0 
or 
d ip 
—+ 1=0 (a) 
d (p 
from equations (1) and (1)', 
dip cos qp, cos ip1 dip' 
d cp COS qp' cos ip d qp' 
From equation (6) 
d*p' . 
d qp' 16 
ON THE DEVIATION OF LIGHT 
this last combined with (b), and (a), we get 
COS Cp . COS <// 
COS (Jo'. COS ip 
which will be satisfied on making <p = V. and 9>' = V''. 
Hence, the deviation is a minimum, when the 
angle of incidence is equal to that of emergence. 
This supposition being made in equations (7) and 
(6), they give 
<p = H“ + d) 
v'=i« 
and these values in equation (1), give, 
sin *(« + *) 
iltTfT" (8) 
we have, therefore, only to measure the deviation 
when a minimum, to find the index of refraction 
of the medium of which the prism is made, sup- 
posing its refracting angle known. 
This furnishes one of the best methods by which 
the refractive power of different substances may be 
found. If the substance be a liquid, we may 
unite two plane glasses, making any angle with 
each other, by means of a little cement placed 
along their edges, and place the liquid between 
them where it will be held in sufficient quantity by 
capillary attraction. AT PLANE SURFACES. 
17 
When the ray is incident at right angles upon 
the first surface, we have, 
qp = 0 
9' — 0 
and from equations (7) and (6), there result, 
d = y — a, 
a — ip’ 
whence 
sin (a -f- £)= w sin a . . . (8)'. 
Deviation at plane surfaces by refraction, will 
be again referred to in a subsequent part of the 
text. 
26. Let MN,MN' (fig. 11), be two plane re- 
flectors, meeting in a line projected in M ; S D, a 
ray incident at the point D, and contained in a 
plane perpendicular to the intersection of the re- 
flectors; it will be deviated at the point D of the first 
reflector, again at the point D' of the second, and 
so on. 
Required the circumstances attending these devia- 
tions. 
Call the first angle of incidence, <jp, 
second, . . , tpz 
third, . . • <J> i 
&c., 
nth, . . . Vn 
In the triangle P D D', the angle at P is equal 18 
ON THE DEVIATION OF LIGHT 
to the inclination of the reflectors, which we will 
call i, and we shall have 
9>i —<?2 =* 
9>2 9>3 =* 
<P3 — 9>4 = * 
<)Pb — 2 9,1-1 = * 
n - 1 <Pn = * 
(*) 
<Pi — <Pn = n — 1 . i 
<Pn= <P 1—n — 1 .i • (9) 
If be any multiple of i, as n— 1 . i> 
(p,—n—l.i — 0, . . . (10) 
or the nth incidence will be perpendicular to the 
reflector, and the ray will consequently return upon 
itself. 
Example ls£. Suppose the angle made by the 
reflectors to be 6°, and the first angle of incidence, 
or 9>i = 60°, required the number of reflections be- 
fore the ray retraces its course. 
These values in equation (10), give, 
60° — n — 1.6° = 0 
or 
ans: n = 11 
Example 2d. The angle of the reflectors being AT PLANE SURFACES. 
19 
15°, and the first angle of incidence 80°, required 
the fourth angle of incidence ; 
These values in equation (9), give 
<p4 =80°.—4—1.15°. 
ans: <p4 = 35° 
If ?>, be not a multiple of i, there will be some 
value for n that will make n — 1. i> greater than 9>j, 
in which case, <Pt —n—1. i will be negative; that is, 
at the nth incidence, the ray will be on the opposite 
side of the perpendicular. It will therefore return, 
but not, as before, by the same path. 
Example 3d. The angle of the reflectors being 
7°, the first angle of incidence 69°, required the 
number of reflections before the ray returns, and 
the first angle of incidence of the returning ray. 
These values in equation (9) reduce it to 
(pn = 69° — n — l . 7° = 76° — In 
If»= 10, 
cpn - 76° — 70° = 6° 
If n= 11, 
(pn — 76° — 77° = — 1 ° 
or the ray begins to return at the eleventh inci- 
dence and the value of the angle of incidence is 1°. 
It is obvious that the angle of incidence of the 
returning ray will increase at every deviation; 
there will, therefore, be some value of the in- 
creased angle which will either be equal to 90° ; 
in which case, the ray will finally be reflected by 20 
OF THE DEVIATION OF RAYS 
one of the reflectors into a direction parallel to 
the other; or greater than 90° when the ray would 
meet the other reflector on being produced back. 
Adding the first two equations in group (a), we 
have 
or 
Vi — <f>3 = 
SS' D=2t 
That is, the angle made by the incident ray and 
the same ray after two reflections, is equal to double 
the angle of the reflectors. It follows, therefore, that 
if the angle of the reflectors be increased or dimin- 
ished by giving motion to one of the reflectors, the 
angular velocity of the reflected ray will be double 
that of the reflector. This is the principle upon 
which reflecting instruments for the measurement 
of angles are constructed. 
Of the Deviation of Rays at Sperical Surfaces. 
27. Let M D O N (fig. 12), be a section of a 
spherical surface separating two media of different 
densities, as air and glass, having its centre at C, 
on the line O C, which will be called the axis of 
the deviating surface ; / D a ray of light, incident 
at D, and D S, the direction of this ray after devi- 
ation, which being produced back will intersect 
the axis at The point O, where the axis 
meets the surface, is called the vertex. Call / D, AT SPHERICAL SURFACES. 
21 
u ; f’ D, vl; CD,r; Off; and the an- 
gle O C D, e. 
In the triangle C D f we have the relation, 
sin cp f—r 
sin d u 
and in the triangle C D 
sin 6 u 
sin <p‘ f—r* 
these combined with, 
we have, 
sin gp — m sin gp', 
. • (II) 
The first of these triangles will also give, 
u8 *= (f— r)8 + r8 -J- 2 (f — r). r. cos 0 
and the second, 
«'2=(/'_r)3 -f r2 +2(/' — r).r,cos d. 
these latter equations by reduction become, 
u1 =/2 — 2 r (/ — r). versin. 6 
u2 ==/'* — 2 r (f' — r). versin . d. 
Denoting the versin. e, by z, and eliminating 22 
SMALL DIRECT PENCIL. 
u and ul, between these equations and equation 
(11), there will result, 
-%r(f—r).z=m(f'—r).'/f‘—2r(f—r).z (12) 
This is a general equation for finding the inter- 
section of deviated rays with the axis. The rela- 
tion between f and f, is somewhat complicated, 
and it is obvious that if f be taken constant, the 
value of f will vary for different values of 6; 
that is to say, if a pencil of rays proceed from a 
point on the axis, they will, after deviation, meet the 
axis in different points, depending upon the dis- 
tance of the point of incidence from the vertex. 
Small Direct Pencil. 
28. A pencil of light having its central ray co- 
incident with the axis of the deviating surface, is 
called a direct 'pencil; and if such a pencil be 
taken very small, the quantity z, in equation (12), 
will be so small that the products of which it is a 
factor, may, without material error, be omitted. 
This will reduce equation (12) to 
(/— r) ./'=»»». (/' — r)./ 
or 
/-sdfc ■ ■ • <■» SMALL DIRECT PENCIL. 
23 
and taking the reciprocal, 
1 m—1 1 
■ ■ ■ (,4) 
If/be constant, or the rays all proceed from the 
same point on the axis,/' will also be constant for 
the same medium, and curvature ; hence, all the 
rays after deviation will meet in the same point 
on the axis. This point is called the focus, and the 
point from which the rays first proceeded, the radi- 
ant; and because of the mutual dependence of 
these points upon each other with respect to their 
position, they are often called conjugate foci. If 
the rays after deviation only meet the axis by be- 
ing produced back, the focus becomes imaginary, 
and is said to be virtual; the radiant is also said 
to be virtual when the rays converge to a point 
before deviation. 
If we suppose the ray deviated at the first, to be 
incident on a second surface M' N', (fig. 13), hav- 
ing a radius r', and at a distance t, from the first, 
measured on the axis, we may regard this ray as 
proceeding originally from /'; and estimating the 
distance from the vertex O' to the point at which 
the ray, after deviation at the second surface, meets 
the axis, we shall have, from equation (14), by call- 
ing this distance /", and the index at the second 
surface m', 
1 m' — 1 1 
~p w7“ +!»'(/'+o * ’ (15 SMALL DIRECT PENCIL. 
The value of (/' +1)9 deduced from (14), 
and substituted in equation (15), gives a direct 
relation between/and/", in functions of r, r', m, 
ml and t. 
For a third surface, as M" N" (fig. 14), we shall, 
in like manner, have 
1 m'—1 1 
/'" “ m" r" m" (/"+ t + ?) ' 
If in this equation, we substitute the value of 
(f" +1 + t'), deduced from the equation derived 
from (14) and (15), by eliminating/', we shall have 
the relation between / and/"', expressed in func- 
tions of r,r', r", m, m', m", t, t!\ and so on, of any 
number of surfaces. 
In the same way (fig. 15), 
1 m"'-1 I 1 ,17v 
“ m"' r" m" (f'"+t+t'+tu) ' 1 } 
28. From equation (13), we get, 
/'+1 — !-/+01 
m — 1 ./+ r 
this substituted in equation (15), at same time 
making ml — which is supposing the ray to SMALL DIRECT PENCIL. 
25 
pass into the first medium after having traversed 
the medium bounded by the two deviating surfaces, 
that equation reduces to, 
1 _ 1 — m m(m — l.f+r) 
/" r mrf+{f.m—l + r)t 
The same process being continued with respect 
to rays deviated at three, four, &c., surfaces, we 
would obviously obtain a series of equations ex- 
pressive of relations between f" and f, flv and 
/, fy and f, &c.; the equations becoming more 
and more complex in proportion as we consider 
a great number of deviating surfaces. They will 
be greatly simplified, however, by neglecting the 
consideration of the thickness, which may be done 
in almost all cases of practice without incurring 
much error. Dropping the thickness, equation 
(18) will become, by retaining m!, 
l m — 1, 1 (m — 1 1 > 
f" m'r' 1 m ( mr mf S' 
the value of substituted in equation (16), gives, 
-I _ ?.-.1 , _L | , 1 , _L\ ( (JO)’ 
mV" + t mV + m' \ mr + mf) 26 
SMALL DIRECT PENCIL. 
and the value of y.777, in equation (17), gives, 
1 m "— 1 
y'"" w'" r';' * 
_l --+-1-) 1 i(i7)' 
m L m r m * ‘ m r m V mr 'fnf'iJi 
and so for additional surfaces. 
If we now suppose the medium between the 
second and third, fourth and fifth, sixth and seventh, 
Sec., deviating surfaces, the same as that in which 
the light moved before the first deviation, we shall 
have the case of a number of refracting media 
bounded by spherical surfaces, situated in a homo- 
geneous medium, such as the atmosphere, for ex- 
ample, and nearly in contact. Hence, 
I „ 1 „„ l . 
m = —;; m = —777; »i = —777-, &c. 
m m m 
and the foregoing equations reduce to, 
l7 = (m_l).|i-—i + ir (19) 
1 vi — 1 , 1 \ / 1 lNli; 
Trr, u—77— 77 ) m — 1 • ( r ) + “T- ( • * (20) 
f m r m ( v r r ' j ) ' 7 
1 / 1 1 \ , /I 1 \ 1 
— =m _ 1 m-1 •(-7-7t)+j(21} 
&c., &c. 
29. Any medium bounded by curved surfaces, SMALL DIRECT PENCIL. 
27 
used for the purpose of deviating light by refraction, 
is called a lens. Equation (19) relates, therefore, to 
the deviation of a small pencil of light by a single 
spherical lens ; f denoting the distance of the 
radiant, and f", that of the focus from the lens. 
Equation (20), relates to the refraction or devi- 
ation by a single lens and a second medium of 
indefinite extent bounded on one side by a sphe- 
rical surface nearly in contact with the lens. 
Equation (21), relates to deviation by two sphe- 
rical lenses close together, f and f" denoting, 
as before, the radiant and focal distances. 
30. If the rays be parallel before the first devia- 
1 
tion, f will be infinite, or -j = 0, and equations 
(19), (20), and (21), will reduce to 
1 /1 1 \ 
jr.-m-l \~—7) 
1 m"—t , 1 r / 1 1 \ 1 
" ~mWr T“»7rL”-|'l7-7iJ 
1 / I 1 \ /I 1 \ 
jnr-m'-1 ■ m-l • 
&c., &c. 
The values of f", f", f"", &c., deduced from 
these equations are called the principal focal dis- 
tances, being the focal distances for parallel rays. 
Denoting these distances by F„, F///,F////,&c., and 
(4-4) ’ (4-4)&c- b4 4 f&c-we sba11 28 
SMALL DIRECT PENCIL. 
have by including equation (14), the following 
table: 
1 _ m—l 
f, m r 
1 m — 1 
3SC ., . - 
Q 
1 _ m"— 1 1 /m— 1 \ 
fiii m!' r" m" V q ) 
1 _ m" — 1 m — 1 
p, ~ 9n ? 
1 _ m"" — 1 1 xm"— 1 m— 1 
— - m"» r"" + \~y~ + “7“ ) 
i __m"”— i jrm—i 
p„„„ ~ e" ? 
&c., &c., &c. 
• • (A) 
An examination of the alternate formulas of the 
above table leads to this result, viz. that the re- 
ciprocal of the principal focal distance of any com- 
bination of lenses, is equal to the sum of the recipro- 
cals of the principal focal distances of the lenses 
taken separately; which may be expressed in a 
general way by the equation, 
r-<T> (82) 
wherein denotes the reciprocal of the SMALL DIRECT PENCIL. 
29 
principal focal distance of any one lens in the 
combination z, that the algebraic sum of these 
is to be taken, and —, the reciprocal for the com- 
bination. 
Substituting the first member of the first equa- 
tion, in group (A), and the first members of the 
alternate equations beginning with the second, for 
their corresponding values in equations (14), (19), 
(21), <fec., we finally obtain, 
hv+ir <23>' 
bbi (S3) 
+7 (24) 
A=i+7 (25) 
Equations (23), (24), and (25), are of a convenient 
form for discussing the circumstances attending the 
deviation of light by refraction through a single lens, 
or a combination of lenses placed close together ; 
and equation (23)', the deviation by reflection at a 
single surface. 
31. The several terms of these equations are the 
reciprocals of elements involved in the discussions 
which are to follow. But, the pencil of light being 
small, the versed sine of the arc O D, (fig. 
16), may be disregarded, and this arc may be con- 30 
SMALL DIRECT PENCIL. 
sidered as coinciding with the tangent at the ver- 
tex O, and as having been described about either 
of the points C, V, or U, as a centre, indifferently, 
hence 
1 1 
d v o ; d u o:: —r : —r, 
f f 
that is, the relative divergence or convergence of the 
incident and deviated pencils will be expressed by 
the reciprocals of the conjugate focal distances/ 
and/'. 
32. The power of a lens is its greater or less 
capacity to deviate the rays that pass through it. 
In equations (23), (24), (25), &c., —, ——, 
F// F/// 
&c., will measure the divergency or conver- 
gency of parallel rays after deviation; and as these 
measures are expressed in functions of the indices of 
_ . 1 ,1 lN 
refraction and or f — — yj, &c„ they will be 
constant for the same media and curvature; and 
hence become terms of comparison for the other 
two terms which enter into the equations to which 
they respectively belong. 
From what has been said, it is apparent that 
—, in equation (22), will measure the degree 
of convergence or divergence of parallel rays after de- 
viation by any combination of spherical lenses what- SMALL DIRECT PENCIL. 
31 
ever, and will consequently be the measure of the 
power of the combination ; and as (—) is the 
measure of the power of any one lens of the com- 
bination, we have this rule for finding the power of 
any system of lenses, viz : Find the power of each 
lens separately, and take the Algebraic sum of the 
whole. 
33. It may be convenient to express the relation 
in equations (23)', (23), (24), &c., by referring to 
the centre of curvature of the deviating surfaces as 
an origin. For this purpose, let O D, (fig. 17), be 
a section of the deviating surface ; denoting the 
distances of the radiant and focal points from the 
centre C, by c and c', respectively, we have by in- 
spection, 
/= r -f c, 
/' 3 r -f c', 
which in equation (13), give, after reduction, 
4-—• • • ■ <»> 
c r c 
and for a second deviating surface whose centre of 
curvature is at a distance t, from that of the first, 
we obtain from equation (26), 
* . . (W) 
c r c — i 32 
SMALL DIRECT PENCIL. 
and for a third, whose centre is at a distance t,' from 
that of the second, 
1 - ”"-11 
r——+?“«_? • -f28' 
&c., See., 
c' being eliminated between (26) and (27), a rela- 
tion between c and c" will result; in like manner, 
c" being made to disappear by means of this de- 
rived equation and equation (28), there will result 
an equation in terms of c'" and c, and so for any 
others. 
Application of the preceding theory to the de- 
viation of light by refraction through the various 
kinds of spherical lenses. 
34. A lens has been defined to be, any medium 
bounded by curved surfaces, used for the purpose 
of deviating light by refraction ; the surfaces are 
generally spherical. 
A, (fig. 18), called a double convex lens, is formed 
by two spherical surfaces, having their centres on 
opposite sides of the lens. When the curvature 
of the two surfaces is the same, the lens is said to 
be equally convex. 
B, is a lens with one of its faces plane, the 
other spherical, and is called a plano-convex lens. 
C, is a double concave lens, having the centres of 
its curved surfaces on opposite sides. 
D, is a plano-concave lens, having one face plane 
and the other concave. SMALL DIRECT PENCIL. 
33 
E, has one face concave, and the other convex 
the convex face having the greatest curvature ; this 
lens is called a meniscus. 
F, like the meniscus, has one face concave and 
the other convex, but the concave face has the great- 
est curvature ; this is called a concavo-convex lens. 
The line U V, containing the centres of the sphe- 
rical surfaces, is called the axis. 
35. A moment’s consideration will show that 
all the circumstances of convergence or divergence, 
attending the deviation of light by any one of these 
lenses, will be made known by equation(23), it be- 
ing only necessary to note the different cases arising 
out of the various combinations of surfaces by which 
the lenses are formed ; these cases depend on the 
signs of the radii. 
Equations (23), (24), and (25), &c., were de- 
duced on the supposition that r is positive when 
the concave side of the surface is turned towards 
incident light; it will, of course, be negative when 
the convex side is turned in the same direction. 
Besides, f was taken positive for a real radiant, or 
when the rays are supposed to diverge from any 
point upon the axis of the lens, before deviation ; 
on the contrary, it will become negative, when the 
rays are received by the deviating surface, in a state 
of convergence to a point behind the lens. The 
signs of f, f", &c., have been taken positive 
when the deviated rays meet the axis on being 
produced back. The foci are then virtual. When 34 
SMALL DIRECT PENCIL. 
the rays meet the axis on the opposite side of the 
lens or lenses, f, f", &c., become negative, and 
will correspond to real foci. 
The several lenses may be described as follows: 
iDouble Convex, . . . — r and -f- r 
Plano-Convex, convex side to incident 
light, . . —r and + r =■ od 
Do. plane side to incident 
light, . -f- r = od and r' 
2 
3 
Meniscus, convex side turned to inci- 
dent light, . r < r, — r, — r 
Same, concave side do. do. r > r -f- r, -j- r 
4 
5 
Double Concave, . . . -j- r, — r' 
Plano-Concave, concave side to incident 
light, . . -f- r, + r = od 
Same, plane side to do do -f- r, = od, and — r 
Concavo-Convex, concave side to inci- 
dent light, r < r' -j- r,-\-r 
' Same, reversed, r>r' —r,—r' 
6 
36. For a double convex lens, (fig. 19), 
' (23)" 
— = — m — 1 ( h —), 
f„ g \ r r / 
and as long as m > 1 we shall have, 
f—v-+7 W" SMALL DIRECT PENCIL. 
35 
For -I- > —r, or / > f;/, f" will be negative ; 
J 
or the focus will be real, and the rays will converge 
after deviation. 
For —— < —or f < f;/, f" will be positive, 
n J 
the focus virtual, and the rays will diverge after de- 
viation. 
If -*-= i, or f„=/; -^=0,orf"= infinity, 
and the rays, after deviation, will be parallel. 
If the rays be received by the lens in a state of 
convergence, y- or /, will be negative, and, 
f" Vf, t f) 
or the focus will always be real. 
If the rays diverge from a point at a distance 
from the lens equal to double the principal focal 
distance, 
f p„ 2p„ 
or 
For all cases of diverging light we have, 
JL<J- 
/" f 36 
SMALL DIRECT PENCIL. 
or, (31), the rays will diverge less after deviation ; 
and for all cases of converging light we shall 
have, 
/" f 
or the rays will converge more after deviation. 
Hence, the effect of a double convex lens is to 
collect the rays. 
The focal distance for this lens in the case of 
diverging rays in functions of m, r, r' and/, is 
f" = f'rr' 
r r' —f (m — l)(r+r')' 
If the lens be supposed of glass, m = ~, nearly, 
and 
fn 2/r r' 
/.(r+r')-2rr ’ 
If the lens be equally convex, r — r', and 
w,___ f r ' 
J /—r ’ 
and if the rays be supposed parallel before devia- 
tion, f = infinity, and 
f" — — r. SMALL DIRECT PENCIL. 
37 
37. A like discussion may be gone through with 
each of the other lenses described. This being 
done, the results will conform to those exhibited in 
the following table. 
Lens. 
Incident pencil. 
1 
T' 
Sign of /". 
Refrac. pencil. 
Diverging 
+ / 
1 1 
+ —?~ 
F/, J 
j/>F„ 
Converges 
Convex 
F 
II 
/<F" 
+ 
Diverges 
less. 
Converging 
-/ 
_i i_ 
F» f 
\r<f 
Converges 
more. 
Diverging 
+/ ! 
>v+7 
{/X/ 
Diverges 
more. 
Concave 
/>PH 
+ 
f Diverges. 
Converging 
U-7 
Converges 
less. 
/< 
/">/ 
A similar table may also be constructed by for- 
mula (24), for a combination of any of the sphe- 
rical lenses taken two and two, and by formula 
(25), for any combination taken three and three, 
and so on. 
In general, it may be inferred from the preceding 
table, that convex lenses tend to collect the inci- 
dent rays, while concave lenses, on the contrary, 
tend to disperse them. 38 
SMALL DIRECT PENCIL. 
38. Differentiating equation (23), we have 
<*/“ dj . 
/"2 f2 ’ 
the upper sign answers to the case where f" and f, 
have different signs, and the lower to that in which 
the signs are the same ; which shows that when the 
conjugate foci are in motion, they move in the same 
direction. 
39. If the lens be a sphere, m! = —, in equation 
m 
(27), and t = 0 ; and eliminating c', by means of 
equation (26), we obtain 
7^2{m—D+L . . . (29). 
c1 mr c 
40. If in equation (14), we make r infinite, we 
get 
1 1 
or 
™f=.f 
which answers to the case of a small pencil devia- 
ted at a plane surface separating two media of dif- 
ferent densities, as air and water. On the suppo- 
sition that the radiant is in the denser medium, as APPLICATION TO THE DEVIATION OF, &c. 
39 
in (fig 20), m becomes —, and this in the prece- 
ding equation gives 
/= *»/'; 
that is, to an eye situated without this medium, the 
distance of the radiant from the deviating surface 
will appear diminished in the ratio of unity to the 
index of refraction of the ray in passing from the 
denser to the other medium. This accounts for the 
apparent elevation above their true positions of 
all bodies beneath the surface of fluids, and for the 
apparent bending of a straight stick when partly 
immersed in water. 
Application to the Deviation of Light by Spherical 
Reflectors. 
41. In reflection, we have only to consider one 
deviating surface. Equation (14) applies here by 
making m = — 1, (17), which reduces it to, 
H-7 ■ • 
But two cases can arise, and these will be dis- 
tinguished by the sign of the radius. The reflec- 
tor may be concave, when r will be positive, or it 40 
APPLICATION TO THE DEVIATION OF 
may be convex, when r will be negative. Equa- 
tion (30), and (fig. 21), relate to the first case, 
which will now be discussed. 
If the incident rays be parallel, ~ =0, and 
1 _ 2 
fT~~^r 
or 
f = — = F 
7 2 
Hence the principal focal distance is equal to half 
radius, and equation (30), reduces to 
hr,~7 • ■ ■ ■ (31> 
11 
As long as—> — or/> f ,f' will be positive, 
J 
which, since the rays are thrown back from the 
deviating surface, will correspond to a real focus, 
and the rays will converge after deviation. 
1 1 
If — < -T or/< f„ f will be negative, and 
J 
there will be a virtual focus, or the rays will di- 
verge after deviation. 
If the radiant be at the centre of curvature, 
f — 2 F/f and 
/' = a P; = r LIGHT BY SPHERICAL REFLECTORS. 
41 
or the radiant and focus coincide. 
For 
/> 2f; or/> r; 
1 . 1 ,, 
f >"277’/ <ri 
or the focus will be between the reflector and cen- 
1 11 11 
tre, and since — — y <—, /, < ~, or /' > f, ; so 
that the focus will be found between the centre 
and principal focus. 
For 
/<2f; or/< r; 
1 , 1 
f <inr’/>r- 
or the focus will be at a greater distance from the 
reflector than the centre. 
When f = f,, y = 0, or the focus will be at an infi- 
nite distance. 
When/< f/,// will be negative, and the rays 
will diverge after deviation. 
If the rays converge before incidence,/ will be 
negative, and equation, (31), becomes 
1=1+1 
f * + f 
Hence,/' will always be positive, or the rays will 
converge after deviation. 42 
APPLICATION TO THE DEVIATION, &c. 
42. By discussing the several cases that will 
arise in attributing different signs to r and/, and 
various values to the latter, we shall find the re- 
sults in the following 
Table. 
Reflector. 
Incident Pencil. 
T7 
Sign of JL 
f 
Reflected Pencil. 
Diverging 
+ / 
i _ i_ 
F,' / 
/> p, j 
+ 
Converges 
/<F, 
/>/ 
Diverges 
less. 
Concave 
+ p, 
Converging 
-/ 
*■. / 
'/'</ 
Converges 
more. 
Diverging 
+/ 
) _1 — Diverges 
I F, f f <f m0re‘ 
Convex 
— F( 
Converging 
. -/ 
_7+7 
/> p, — Diverges, 
-j- Converges 
/<r /■ >/ less. 
Hence, in general, concave reflectors tend to col- 
lect the rays of light, and convex to disperse them. 
43. Differentiating equation (31), we have 
df' d/ . 
y's BS3± ya » SPHERICAL ABERRATION. 
43 
The upper sign corresponds to the case when f 
and f have the same signs. Hence the conju- 
gate foci of spherical reflectors will move in op- 
posite directions. 
44. Equation (30), by making r infinite, re- 
duces to 
f'~ f 
or 
Which shows, that in all cases of deviation of 
a pencil by a plane reflector, the divergence or con- 
vergence will not be altered ; and if the rays diverge 
before deviation, they will appear after deviation to 
proceed from a point as far behind the reflector as 
the actual radiant is in front; but if they converge 
before deviation, they will be brought to a focus as 
far in front as the virtual radiant is behind the re- 
flector. 
Spherical Aberration. 
45. Thus far the discussion has been conducted 
upon the supposition that the pencil is very small, 
and that z, the versed-sine of the arc 6, included be- 
tween the axis and the extreme rays of the pencil, 
is so small, that all the products of which it is a 
factor may be neglected. If, however, z be retained 44 
SPHERICAL ABERRATION. 
and we find the value of /', in equation (12), by 
Maclaurin’s Theorem, we shall have 
Wherein (/')0 is the value of /', as given by equa- 
tion (13). 
In questions of practice, z is so small that the 
higher powers than the first may be neglected 
without impairing greatly the accuracy of the ap- 
proximation as given by the above series. Retain- 
ing, then, only the first power of z, we shall have 
r-w>-+(£);: 
df' 
To find jj, we resume equation (12), and dif- 
ferentiate it, regarding/' and z as variable. 
(f—r) Vf>2 -2 r(f'—r)z~m{f — r) Vf*-2r(f—r)z. 
(/— r) • If -df —r z.d.f — r(f — r)dz] _ 
Vj^--2r(/'-r)z 
, . r. (f — r) d r 
”J/'(r-2r(/-r)2-.(/-r)-j==== 
Making z = 0,/' assumes the limiting value (/')Q SPHERICAL ABERRATION. 
45 
as given by equation (13), and we have, 
, „ % (/ )o df‘ — r ((/')„ — r)dz 
(f-r). (7)7 - 
772-jT • Cl j Of Z» 
ir 7r, r((./)•—r) , > 
</-r) (7)7 " 
, ,r, m.r.((/')0-r)(/-r), 
mf.df— (iar. 
w.r((y')o—r) ( f—r) d z 
*/• 4/ - (/- r) if = ~ “ 
— r((/')°~r) (f~r)-dz 
(/')o ' 
= r. (f—r) ((f)0 — r) 1 
m — 1 ./+r V/ 
but from equation (13), we have 
(. r = _zy~rl 
w—1 .f -\-r m—l/+r 
hence, 
dz (/)o r*(/ (/% )’ ' * ' * (32) 
and 
/' “ (/)<» + (/')—o r’(Kf (y')0 )’* * * • ' 
46. This equation relates to a single surface ; if 46 
SPHERICAL ABERRATION. 
we pass to a second, and limit the investigation to 
the case where ?n = —which supposes the light to 
pass to the same medium from which it entered the 
first deviating surface, the result will appertain to 
a single lens. 
In equation (15), we have f" a fnnction of f ; 
the latter, in equation (12), is a function of z ; and 
calling z'f the versed-line of the arc & at the se- 
cond deviating surface, included between the axis 
and the extreme ray deviated at the first surface, 
f" will also obviously depend upon z', which, to- 
gether with z, was neglected in finding equation 
(19). Hence we shall have /", a function of 
(z, z'). By the general development of a function of 
two variables, limiting the series to the terms in- 
volving the first powers of the variables, for the 
reason given in art. 45. 
+ + ' ' ' (34)' 
To obtain the first partial differential co-efficient, 
we have 
rdf\ df" df 
\dz Jo df dz ’ 
df" 
and to obtain jjr, it will be sufficient to take equa- SPHERICAL ABERRATION. 
47 
tion (15). Neglecting t, and making m' — we 
find, 
df" 
df (/') o2 * 
and is given by equation (32). 
The second partial differential co-efficient, is 
found at once from equation (32), by changing / 
into/', r into r', z into z!, and m into — which will 
u 7 7 m, 
give, 
(1£)r((/'W)1^-wS- 
These several values being substituted in equa- 
tion (34), give 
r-tr) \m{f")o2 • rm—VY- —\z 
f (/)o+(/^2 ((/)„-r). (/-)o } 
+ (7w'(^j7-T7V>' ■ (35) 
If the arcs d and & be taken so small that z 
and z', their versed-sines, may be neglected, this 
equation reduces to 
/''-(/") o 48 
SPHERICAL ABERRATION. 
as in equation (19). Subtracting this last equa- 
tion from (35), we have 
r" i f'\ _mif")°~ (f\ rl / m 1 A . _ 
J if) o- (/v .(Sb r. - 
+ • <36) 
Now/", is the focal distance of the rays which 
are incident at the distances d, 6', from the common 
axis of the deviating surfaces, and (/")D, the focal 
distance of those incident very near; hence the 
difference, or /" — (/")0, will be the length of 
that portion of the axis upon which will be situated 
the foci of the rays between the boundary of the 
pencil and its central ray, supposed coincident 
with the axis. This wandering of the deviated 
rays from a single focus, thus shown to exist in the 
case of spherical surfaces, is called spherical 
aberration. 
Let O, be the focus corresponding to (/")Q, in 
figure (22); O', the focus corresponding to/". The 
distance O O' is called the longitudinal aberration; 
and if a perpendicular to the axis be drawn through 
O and produced till it meet the’deviated ray through 
O' in b, O b, is called the lateral aberration. Call- 
ing the longitudinal aberration a, we shall have, 
171 if )o2 . / ( f'\ r\2 f 171 * . z 
if)o2 ((/)o v/ (/')o y 
• <37> SPHERICAL ABERRATION. 
49 
C, being the centre of curvature of the second 
deviating surface M M7, DP = r sin 6, is called the 
radius of the aperture; and if z and z7 be equal, 
or nearly so, it follows from equation (37), that the 
longitudinal aberration for a given value of f in a 
given lens, will vary as the square of the radius of 
the aperture. 
47. Calling the lateral aberration b, the longi- 
tudinal a, as before, the similar triangles of the 
figure give 
b r sin 6 
a f"+z’ 
but z maybe neglected as compared with/", with- 
in ordinary limits, hence 
, a . r . sin 6 
h—~r~ (38) 
and as a varies as (r sin of, the lateral aberration 
will vary as the cube of the radius of the aperture. 
48. Resuming equation (33), and making 
m = — 1, it reduces, calling a the longitudinal 
aberration, to 50 
SPHERICAL ABERRATION. 
In figure (23), O' and O, being the foci corres- 
ponding to the values /' and (/')„, we shall have 
by inspection, denoting the lateral aberration by b, 
b rsin0 
a ~ f ' — z’ 
or 
a . r sin 0 
b=—f— 
from this, and equation (39), we infer, that in re- 
flection at spherical surfaces as in refraction, the 
lateral aberration varies as the cube, and the lon- 
gitudinal as the square of the radius of the aper- 
ture. 
49. Since the rays deviated at spherical sur- 
faces are not brought to a single focus, it becomes 
a matter of some interest to ascertain the magni- 
tude and position of the least space that will con- 
tain them all: this space is called the circle of least 
diffusion. 
Let O (fig. 24), be the focus corresponding to the 
value and O' to the value f". If a ray 
be deviated on the side of the axis opposite 
to 6, and at a distance less than n, it will meet 
the axis at O ", between O and O', and intersect 
the extreme ray passing through O' produced at y. 
Another ray deviated at a distance 0'", greater 
than 0", will belong to a focus between O' and O", SPHERICAL ABERRATION. 
51 
but may meet the extreme ray produced to the 
right or left of y, making the length of xy greater 
or less than that corresponding to e"\ hence if a 
value for xy be found answering to a maximum, 
all the rays will necessarily pass through the cir- 
cular space having this value for radius. 
Call a the longitudinal aberration for o; a!' that 
for d"; b and b" the corresponding lateral aberra- 
tions ; x, the distance of the circle of least diffu- 
sion from O', and y its radius. Then?/ will be a 
maximum when x is so. 
Equation (38), gives 
b r sin 6 
I 
b" rsin 0” 
j*~r » 
a J 
by division, 
b a" sin 6 
b" a sin 6" * 
x a 
V b ' 
a — a” — x a' 
y = IbT' 
by division, 
a — a" — x _ a" b sin d 
z = ITh “ sin^7’ 52 
SPHERICAL ABERRATION 
hence, 
„ sin 6 -f sin 6" 
a a —X. : 277 j 
sin a 
but from the relation between the longitudinal 
aberration and radius of aperture, we have 
a!' _ sin2 6" 
a sin2 0 
or 
„ sin20—sin2 6" sin 6 -f- sin 6" 
a a = a . ;——2 =* : 277 ■ 
sin2 o sin .6 
hence, 
sin 6" (sin 6 — sin 6") 
x~a • a • 
sin-2 6 
dx sin 6. cos 6" — 2 sin 6". cos 6” 
HW = 
or 
sin 6" — \ sin 0, 
hence, 
® = a; 
and 
y™ ib- 
Again, 
d'1 x sin &' -f- 2 
dW~ sin 0 ""» 
y is, therefore, a maximum. SPHERICAL ABERRATION. 
53 
50. In equation (33), which answers to one sur- 
face, there will be no aberration when either of the 
factors in the last term of the second member is 
equal to zero. Taking the first factor, 
( / )o r = 0, 
we obtain from equation (13), 
= (/')« : 
and the second factor, 
m 1 n 
7"~T7T“ ’ 
replacing by its value given in equation (14), 
and reducing, we have 
f=r(m-f 1) 
If r be positive, the deviating surface will have 
its concavity turned towards incident light; if ne- 
gative, its convexity. In the first case, the incident 
rays will diverge, and if the first factor be zero, 
they will diverge from the centre of curvature and 
will, of course, undergo no deviation ; but if the 
second factor reduce to zero, the rays will diverge 
from a point beyond the centre and distant from it 
equal to m r. In the second case, viz., where r is 54 
SPHERICAL ABERRATION. 
negative, the incident rays will converge to the cen- 
tre of curvature when the first factor is zero, or to 
a point beyond the centre, when the second factor 
is zero. 
In the case of reflection, m = — 1, and the second 
factor cannot reduce to zero. 
Generally, then, every spherical refracting sur- 
face has two points on its axis, so related that all 
rays proceeding from or to one of them, will 
after deviation, proceed from or to the other. 
These points have been called aplanatic foci; the 
first being called the aplanatic focus for incident, 
the other for refracted rays. The distance of the 
first from the surface is given by 
/=(w + l).r 
and that of the other by 
m + I r 
m 
51. A similar analysis might be made of equa- 
tion (37), but it would lead to investigations too 
long and difficult for an elementary work like this. 
The total amount of longitudinal aberration due to 
the action of two surfaces may be readily found, 
however, from this equation. 
Example 1 st. Required the longitudinal aberra- SPHERICAL ABERRATION. 
55 
tion due to the action of a glass concavo-convex 
lens on parallel or solar rays, wherein, 
m = f, 
r = 1, 
r =•• f, 
In equations (14) and (19), y, being = 0, 
/= 3 
f" — 5 
hence from equation (37), 
C 3 5.2 1 10.2 /2 1 K ) 430 
a~ ) 2 ' 3.2 2 '3 + 3.*(3 3 5 / ) 81 Z 
z and z' being supposed equal, and if 6 be taken 2°. 
a — — '003 
Example 2d. Required the longitudinal aberra- 
tion for parallel rays in a double concave lens, 
wherein, 
r = r' — 5 
m = f 
z = .0006 56 
SPHERICAL ABERRATION. 
1 
y, being zero as before, and r' being negative, 
equations (14) and (19), give 
/' = 15 
/" = 5 
and equation (37), 
a = —. 023. 
Example 3d. Required the aberration for paral- 
lel rays in the case of a double convex lens, hav- 
ing as before, 
r = r' = 5 
m — f 
z— .0006 
r, being negative, and r' positive, 
/' = —15 
/" = - 5 
and 
a = .oi 
In each of these examples, the aberration being 
of a sign contrary to that off", as given by equation 
(19), its tendency is to shorten the focal distance. 
And this is true for all single spherical lenses 
constructed of any known medium, no matter what OBLIQUE PENCILS. 
57 
the degree of curvature. The destruction of spheri- 
cal aberration in a single lens, for parallel rays, is, 
therefore, impossible, though by the use of two 
lenses placed close together, it may be effected in a 
variety of ways. Such combinations are said to be 
apian atic. 
Oblique Pencils. 
52. Heretofore the radiant has been taken oil 
the common axis of the deviating surfaces, and the 
axis of the pencil supposed to coincide with the 
same line; the axis or Central ray of the pencil be- 
ing, in this position, normal to all the surfaces, has 
either undergone no deviation in passing from one 
medium to another, or been driven back upon it- 
self when not permitted to enter the media before 
it; the other rays of the pencil have, moreover, 
been deviated so as to intersect the axis when not 
rendered parallel to it, and consequently to have 
their foci upon that line. 
But when the radiant is taken off the common 
axis, the rays of the pencil, including its central 
one, will in general be oblique to the surfaces, and 
a new state of things will arise. The pencil is 
said to be oblique, and it is now proposed to in- 
vestigate the circumstances attending its deviation. 58 
OBLIQUE PENCIL 
Oblique Pencil through the Optical Centre. 
53. We have seen, article (20), that a ray un- 
dergoes no deviation when it passes through a me- 
dium bounded by two parallel planes. If, then, in 
the new position of the radiant, we suppose the 
pencil to increase indefinitely, there may always 
be found one ray that will enter and leave the lens 
at points, where tangent planes to its two surfaces 
are parallel. This ray being taken as the axis of 
a very small pencil proceeding from the assumed 
radiant, will contain the focus of the others, the dis- 
tance ofvvhich from the lens, in very moderate obli- 
quities, will be measured by/", given in equation 
(19). ' 
To find where the ray referred to, after undergo- 
ing one deviation, intersects the axis of the sur- 
faces, let figure 25 represent a section of a concavo- 
convex, and figure 26 that of a double convex lens. 
Call the distance between the surfaces, measured 
on the common axis, t, and let e be the distance 
from the first surface to where the line, joining the 
points at which any two tangents are parallel, cuts 
the axis. Then since the radii at these points are 
parallel, the similar triangles of figure 25 give, 
r r 
r — e r' — t~-e 
hence, 
rt 
a * ——- : 
T —r THROUGH THE CENTRE. 
59 
and of figure 26, 
> 
r r 
T C T t -j- C 
hence, 
r t 
e —; : 
r -f r 
or generally 
<=-rh-' ■ ■ ■ (40> 
the upper sign answering to the case where the 
radii have the same sign. 
This value of e, being constant for the same 
lens, it follows that all rays which emerge from 
a lens parallel to their original directions, after 
deviation at the first surface, proceed in directions 
having a common point on the axis. This point 
is called the optical centre of the lens, and may lie 
between the surfaces or not, depending upon the 
form of the lens. 
If we suppose but one surface, that there may 
be no deviation, e must be equal to r ; and the axis 
of the pencil must pass through the centre of cur- 
vature. 
Oblique Pencil not passing through the Centre. 
54. Now let the radiant be assumed at pleasure, 
either on or off the axis, and a ray not passing 60 
OBLIQUE PENCIL NOT PASSING 
through the optical centre be taken as the axis of 
a small pencil; this ray will, of course, undergo 
deviation as well as all others of the pencil, and 
the circumstances attending the directions of the 
deviated rays will be different from those before 
considered. 
Let O (fig, 27), be the position of the radiant, 
O D, OD', two contiguous rays in a plane passing 
through the radiant and the centre of curvature C, 
and O' D, O' D', the directions of these rays after 
deviation. Putting O D = u ; O D'= u + du; DD' 
= m'; D' O'=u' + du\ and CD=CD' 
=? r ; we shall have 
d u = d s. sin q>, 
du =ds . sin cp', 
whence, 
d u sin cp 
d u sin cp' 
or 
du— indu*= 0 , , . . (40) 
Joining the points O and O' with C, and calling 
O C, k, and O' C, k!, the triangles DOC and 
C O' D will give, 
k- — u1 + r2 — 2 r u. cos cp 
k'* =» t<'‘2 4 — 2 r u cos <p'; THROUGH THE CENTRE. 
61 
and since the position of O', is determined by the 
intersection of two contiguous rays, k' will re- 
main constant in passing from D to D'; hence, 
0 — u.du — r . du. cos cp r u . sin cp. d cp 
0 = u .du — r .du . cos cp -\-ru sin cp' .dcp' 
or 
u—rcos cp du 
u « — . —— 
r . sm <p dcp 
, u — r cos cp d u' 
u ;— . —----- 
r. sin cp d cp 
dividing the second by the first, 
u u—rcoscp' sin cp m.du d cp 
u u — r cos cp ' sin cp'' d u ’ m.d cp' 
1YI • d til 
replacing — , by its value in equation (40), 
Cl 71 
d v 
and —-f deduced from equation (1), we get, 
m.d <f> 
u u—rcoscp' sing> cos cp' 
u u—rcosg> sing/ cos cp 
whence 
, ur .cos cp tan. 9 tAn. 
u «■ ; . . (453) 
«.tan cp—(«—rcos g>)tang> 62 
OBLIQUE PENCIL NOT PASSING 
The position of the radiant being given, this 
equation will determine that of the corresponding 
conjugate focus for those rays of an indefinitely 
small pencil which are contained in a plane 
passing through the radiant and centre of curva- 
ture. If that plane containing the axis of the pen- 
cil, and which is called the principal plane, 
be revolved about the line O C, it will cut in 
succession different sets of rays from the pen- 
cil, whose foci will also be determined by 
equation (42), and all of which will intersect 
O C. These foci will evidently lie in a small 
curve described by the point O', in its motion 
about O C ; and the plane of this curve, called the 
secondary plane, will be perpendicular to the prin- 
cipal plane. Hence, considering the small arc as 
a right line, we may infer that in any small oblique 
pencil, all the rays, after deviation, will pass through 
two lines in planes at right angles to each other. 
These are called focal lines, and their property of 
intersecting all the deviated rays, astigmatism. 
55. To ascertain the form which this small de- 
viated pencil takes, let the transverse section of 
the pencil, before deviation, be a circle whose 
diameter is z. An oblique section by the devia- 
ting surface will be an ellipse whose axes are z, 
and z sec. <p, and the deviated pencil will assume a 
conoidal shape, the sections of which, by the princi- 
pal plane and a secondary plane containing the axis 
of the conoid, will be triangular, having their ver- THROUGH THE CENTRE. 
63 
tices in the focal lines, and their bases in the deviat- 
ing surface. The transversal section will in general 
be an ellipse. Suppose one of the latter sections 
to be made at a distance x, (fig. 28), from the de- 
viating surface and parallel to it; and let v, be the 
length of the axis D B, between the same surface 
and the focal line in the principal plane ; then call- 
ing l, the diameter of the transversal section in 
this latter plane, and h that in the secondary plane, 
we have, 
u : u — a:: : sec q>: l 
v: v — x:: ; h 
or 
I _- . A. sec gt>. . . (43) 
h = (44) 
as x increases, l and h will decrease. 
When 
X = U, 
1=0 
V — u 
h = — —- K 
v 
or the ellipse becomes the secondary focal line« 64 
OBLIQUE PENCIL NOT PASSING 
If x be greater than u', l will increase, and h will 
still decrease as long as x < v; l and h will, there- 
fore, be equal at some point where the section will 
become a circle. To find this point we have, 
U—X . v—x . 
;— . *. sec qo = . * 
« V 
or 
U (1 4" cos 9) 
X — ; 1 
u , 
1-1 . COS qt> 
V 
which will give the position of the circle ; and its 
diameter will be given by putting this value for x, 
in equation (43) or (44). 
Making this substitution and reducing, we get 
v-\-u COS Cf> 
The circle of which this is the diameter, is called 
the circle of least confusion, because within it the 
rays approach most nearly to convergence. 
If x continue to increase and become equal to v, 
A — 0 
1 = 1. sec <p'. ——7—^, 
and the ellipse becomes a line equal to the primary THROUGH THE CENTRE. 
65 
focal line, the length of which is, therefore, known. 
Beyond this, both / and h increase indefinitely. 
To pursue the oblique pencil through a second 
deviating surface, would lead to difficulties which 
we will avoid, our principal object having been 
attained, viz : to show the want of accurate conver- 
gence in any system of rays which includes pencils 
of considerable obliquity, and to prepare ns for un- 
derstanding the action of deviating surfaces in the 
formation of optical images. 
56. Equation (42), may be put under the form 
, tan cp , 
u r . cos cp , 
, tan cp 
u . . . (40) 
tan cp 
u. ;—(u—rcosqp) 
tan cp v ' 
The radiant being supposed on the axis, or <p = 0 ; 
we have u' =/'; u =/, and 
sin cp tan cp 
—: r— r= m 
sm cp . tan cp 
hence equation (45), reduces to 
r = rmf 
(m-l)f-fr ’ 
the same as equation (13). 66 
OPTICAL IMAGES. 
If the pencil be deviated by reflection, we 
shall have 
<f> = <p' 
tan (p 
= m = — 1 
tan cp 
Substituting in (45), and taking the reciprocals 
and reducing, we get 
J I 2 
__ _j 
u u r cos qp 
Taking the radiant on the axis, vl will become f\ 
and u, f; and this last equation reduces to 
i+i=i 
f + f r 
same as equation (30). 
Optical Images. 
57. The surface of every luminous body is made 
up of a vast number of radiants, from each of 
which, rays of light are supposed to proceed in all 
directions not intercepted by the body itself. These 
rays, of course, cross each other; and if any devia- 
ting surface be presented, it becomes the common 
base of a multitude of pencils, whose vertices are OPTICAL IMAGES. 
67 
the radiants constituting the surface of the body. 
If the body be so situated with respect to the de- 
viating surface, that each pencil shall have one of 
its rays passing through the optical centre, it is ob- 
vious from article (53), that those rays of each pen- 
cil in the immediate vicinity of this one, assumed 
as an axis, will, in the case of moderate obliquities, 
be brought to a focus after deviation ; and hence, 
for each radiant, there will be a corresponding con- 
jugate. These conjugate foci make up a second 
luminous surface, from which rays will proceed as 
from the original body ; and this surface is called 
the image of the body, because to an eye so situa- 
ted as to receive the rays as they diverge from it, 
the object, though often modified in shape and size, 
will seem to occupy the position of the new sur- 
face. 
The formation of an optical image consists, 
therefore, in so deviating the rays of light which 
proceed from any object, that the whole or a part of 
them shall unite and again proceed from some new 
position, in a manner similar to that of their lea- 
ving the object. In general, but a part of the rays 
will be deviated to satisfy these conditions, for 
those remote from the axis of each pencil cannot, 
in consequence of aberration and astigmatism, be 
brought to accurate convergence. 
58. To ascertain the relation, between an ob- 
ject and its image, let us suppose the deviation to 
be produced by a lens, so thin that its thickness 
may be neglected. The optical centre, in this 68 
OPTICAL IMAGES. 
case, may be taken as the origin of co-ordinates. 
Denoting by l, the distance from this point to any 
assumed point in the object, and writing this quan- 
tity for/, in equation (23), we get 
..... (46) 
i+f 
Let the object (fig. 29), be a right line, perpen- 
dicular to the axis of the lens. Call e, the angle 
included between the axis of any oblique pencil 
and the axis of the lens. When the pencil be- 
comes direct, 6 will be zero, and l will equal/. 
But, generally, we have 
cos 6 ’ 
this in equation (46), reduces it to 
r-—~— • • • m 
1 -f- cos 6 
which is the polar equation of a conic section.. 
Comparing it with 
A(l— «») 
1 -f e cos v * OPTICAL IMAGES. 
(See Davies’ An. Geom., Book IV. Sch, 5th), 
we get 
f"=r, 
P =:A(1_€8) = ® . . (48), 
A 
• m 
8 = v. 
• B 
For the same lens, p/7 is constant; its value —, 
A 
in equation (48), which is the radius of curvature 
at the vertex, is also constant. 
From equation (49), it is easily seen that the 
curve will be the arc of a circle, ellipse, parabola, 
hyperbola, or a right line, one of the varieties of 
the hyberbola, according as 
F 
/“O’ 
F 
111 S 1 
f< ' 
£«.i 
/ ’ 
/>i. 
cc. 
f 70 
OPTICAL IMAGES. 
or according as the distance of the object is infi- 
nite ; greater than the principal focal distance of 
the lens ; equal to this distance ; less than this dis- 
tance ; or zero. 
If, now, the section represented in figure 29 
be supposed to revolve about the axis of the lens, 
the object will generate a plane, and the image a 
curved surface whose nature will depend upon the 
distance of the object. 
We have seen, article (34), that a positive value 
for f", answers to an imaginary, and a negative 
value, to a real focus ; so, if the points of the image 
be indicated by positive values for f", the image 
will be imaginary ; if by negative values, real. f/; 
for a concave lens is positive, and equation (47), 
answers to this case. Fn for a convex lens is ne- 
gative, and equation (47) becomes 
j*" — £// 
1 yt QOS d 
and the image will always be real as long as 
cos e < 1 
or 
>P 
cos 6 " OPTICAL IMAGES. 
71 
That is, if from the optical centre (fig. 30), with a 
radius equal to the principal focal distance, we de- 
scribe the arc of a circle, and this arc cut the object, 
all that part of the object included between the points 
of intersection a and a', will have no image, or the 
image will be virtual, while the parts without 
these limits will have real images ; if the dis- 
tance of the object exceed that of the principal 
focus, the whole image will be real. 
In general then, in a concave lens, the image is 
always imaginary or virtual, and in a convex lens, 
real, as long as the distance of the object is greater 
than the principal focal distance ; within that limit 
it is also imaginary. 
59. If the linear dimensions of the object be 
small, as compared with its distance from the opti- 
cal centre, cos e, in equation (47), may be taken 
equal to unity; making this supposition and redu- 
cing, we get for a convex lens 
r = -f-rf=--’V • • (50) 
J ru i 2J1 
f 
this value of f", being constant for the same po- 
sition of the object, the image will be a circle ; 
and since the axes of all the pencils intersect at the 
optical centre, we may, without material error, as- 
sume every object, in the case supposed, a small 
arc of a circle having the same centre as the circu- 72 
OPTICAL IMAGES. 
lar image. The object and image will, therefore, 
be similar, and if any linear dimension of the 
former be represented by 8, and the corresponding 
dimension of the latter by <?', we shall have (fig. 31), 
<5' _ f" 
8 = / ’ 
substituting this value in equation (50), it re- 
duces to 
— jA- • • • (61) 
8 f—*„ 
If the distance of the object from the lens be 
equal to twice the principal focal distance, 
7 ’ 
or the object and image will be of the same size* 
If the distance of the object exceed double the 
principal focal distance, the image will be less than 
the object; if its distance be less than double the 
principal focal distance, the image will be greater 
than the object. 
It also follows from the fact just alluded to, viz : 
the intersection of the axes of the several pencils 
at the optical centre, that the real image will 
always be inverted with respect to the object. If the 
rays which proceed from this image be again de- 
viated so as to form a second image, it will be in- OPTICAL IMAGES. 
73 
verted with respect to the first image, but erect as 
regards the object. 
60. If an image be formed by deviation at a sin- 
gle surface, its points will (fig. 32), be readily found 
by means of equation (26). The optical centre, in 
this case, being at the centre of curvature. 
Writing/ for c, and f for c', that equation be- 
comes 
I m — 1 | m 
7 = ~^~+7’ 
making/= co 
_L = _^—1 =7 ... (i) 
f r v ' 
hence, 
1 _ 1 m 
7“7+7; 
or 
f,= f_ = 
/+ *» *, j , mr, 
f 
For an oblique pencil passing through the opti- 
cal centre, we have, on the supposition that the 74 
OPTICAL IMAGES. 
object is a right line perpendicular to the axis of 
the surface, 
F 
/' = 
, .mF, 
1 -f y-" C0S 0- 
wherein —y 6 = Z, as in art. (58), and if the image 
be formed by reflection, m— — 1; hence 
/' = • • • (52) 
1 "byT COS 6 
since f, becomes negative, equation ([k). This 
is a polar equation of a conic section, the nature of 
which will result from the relation of f/ to f It will 
be an ellipse, parabola, or hyperbola, according as 
/>v,/=f ; or/0,- 
The axis of the pencils crossing at the centre C, 
(fig. 33), the image when real will be inverted as re- 
spects the object, and when the object is small, we 
shall have from the similar triangles of the figure, the 
linear dimensions of the object to those of the image 
as the distance of the object from the centre, to that 
of the image from the same point. OPTICAL IMAGES. 
75 
We get the point in which the image cuts the 
axis by making 
6 =0, 
or 
/' = ■ -(53) 
l+y 1+4 
/ F, 
This value of f' being negative, the image will 
be found on the left of the centre, the distance / 
having been taken positive to the right. As long 
as/is positive, the image will lie between the cen- 
tre and reflector; f will be less than f and the 
image, consequently, less than the object. When 
/ is zero, /' will also equal zero, and the object 
and image will be equal and occupy the centre. 
When/becomes negative, or the object passes be- 
tween the centre and reflector, /' will be positive 
as long as/<f,, and the image will pass without; 
f will be greater than / or the image will 
be greater than the object. When f being 
still negative, is equal to f/? or the object is in 
the principal focus, the image will be infinitely dis- 
tant. The object still approaching the reflector,/7 
will be greater than f,;/' becomes negative again 
and the image will approach the reflector from be- 
hind it, and will be greater than the object till 
f— 2 f, or the object be in contact with the reflec- 
tor, when f will equal /, and the image and 
object be of the same size. If now, the object CAUSTICS. 
pass the deviating surface, and reflection take 
place at its other face, we shall have the case of a 
convex reflector, and equation (53) becomes, 
/'- = • • • (64) 
1—5 1-i 
/ F, 
This value of is always negative, greater than 
— p, and less than — 2 Fy, for all values of /, be- 
tween — 2 f7 and infinity, or for any position of 
the object from the surface of the reflector to a 
point infinitely distant in front. In the latter posi- 
tion,/' is equal to — f7, or the image is in the prin- 
cipal focus. It follows also, that the image, which 
will always be virtual, will be elliptical, erect, and 
smaller than the object. 
Caustics. 
61. In discussing the deviation of oblique pen- 
cils at spherical surfaces, we found in equation (42), 
a general value for the length of any deviated ray 
in the principal plane, included between the devia- 
ting surface and the point in which it is intersected 
by the ray next in order. This value depends 
upon u, <?, and r, or the position of the radiant, the 
angle of incidence, and radius of the surface; and 
for any assumed radiant, it is obvious from the na- CAUSTICS. 
77 
ture of the function, that the points of intersection 
of the consecutive rays will not all have the same 
position, but will trace out a curve to which the 
deviated rays will be tangent. This curve is called 
a caustic. 
To illustrate, let us suppose that the radiant 
is infinitely distant, or the rays parallel; equation 
(42) will reduce to 
, r cos 9' . tan 9 
u = .... (55) 
tan 9 — tan 9 ' 
or 
, r cos2 9'. sin 9 
u *=—— —- .... (56). 
sm(9 —9) 
and to construct this, let A B (fig. 34), be an inci- 
dent, and B O, the corresponding deviated ray ; C, 
the centre of curvature. Draw C n perpendicular 
to B O, s perpendicular to the radius B C, and 
s O parallel to A B. O, will be a point in the 
caustic : for 
b n = r cos <p' 
B S*B)i cos 9' = r cos2 9' 
and in the triangle B s O, we have 
r cos2 9'. sin 9 , 
B O = : jr~= «. 
sin (9 — 9 ) 78 
CAUSTICS. 
If we suppose <p = 90°, or the incident ray tan- 
gent to the deviating surface, equation (56), re- 
duces to 
u —r cos go', 
draw C O' perpendicular to the deviated rayB' O', 
and O', will be the corresponding point in the caus- 
tic, and is evidently one of the limits of the 
curve. 
To find the other limit which is on the axis 
A" C, and which answers to w = 0, we cannot use 
equation (56), since it assumes the form of inde- 
termination ; equation (55) will, however, give the 
required point. Putting it under the form, 
, r cos go' 
y — - 
tan cp1 
tan go 
and recollecting that in the case supposed, 
tan go' sin go' 1 
tan cp sin go m 
it reduces to, 
, m r 
u = — . 
m — 1 CAUSTICS. 
79 
and taking B" O", equal to this value of u', O" 
O O7, will be that branch of the caustic resulting 
from deviation between the limits B7 and B77. The 
same thing will, of course, take place on the oppo- 
site side of the axis, and the caustic will have two 
branches uniting on the axis at O77. 
If in equation (42), 
u tan cp — u tan cp -f- r cos cp tan cp1 — 0 
or 
r cos cp. tan q> sin <p 
u —rcos- cp. —— 
tan cp — tan cp sin (9 — cp ) 
n! will be infinite, and the caustic will be divided 
into two branches, lying on opposite sides of the 
deviating surface ; for, any two values of 9, the 
one greater, and the other less than that which 
will satisfy this equation, will give u' opposite 
signs. One of these branches will, of course, be 
virtual, and the deviated ray answering to as 
given by the above equation when u is assumed, 
will be an asymptote to both branches. 
62. Taking, in the general equation (42), m——1, 
or 9 =—9> the resulting formula will belong to the CAUSTICS. 
caustic produced by reflection. Making the re- 
duction, we get, 
, wr cos 9 
u = . . . (5/) 
2 u—r cos 9 
or 
, r. cos 9 
u = . . . (oo) 
r cos 9 
““““*1 • 
u 
The rays being supposed parallel, we have 
u' = £ r cos 9. 
the construction of which is very easy. Let A B 
(fig. 35), be any incident, and B n the correspond- 
ing reflected ray ; draw C n perpendicular to the 
latter, and take B O equal to one half B n, or 
one fourth the chord BS; O will be a point in the 
caustic. Making 9> = 90°, u will equal zero; 
making 9 = 0, u' will equal one half radius, and the 
caustic will commence at B\ and terminate in the 
principal focus. 
If deviation take place at the convex surface of 
the reflector, r will be negative, and 
u‘ as — cos9; CAUSTICS. 
81 
the caustic will be virtual and similar in all re- 
spects to that above. 
With C as a centre, and a radius equal to one 
fourth the diameter of the reflector, describe the cir- 
cle O" Y; with t, the middle of Y B, as a centre 
and radius Yt, describe the circle V O B, which will 
pass through O ; the arc Y O" will be equal to the 
arc Y O, and the caustic will be an epicycloid de- 
scribed by the motion of the circle V O B on O" V. 
For, join V O and t O ; B O being equal to half 
B n, and B V half B C, Y O and C n are parallel; 
V O B is a right angle, and O is in the circumfer- 
ence of which V B is the diameter. The angle 
CBO = ABC = BC B'-i V t O ; butCY=2*Y, 
hence V 0"= V O. 
If the radiant be taken at the extremity of the 
diameter, u (fig. 36), will be equal to 2 r cos <r, 
and this substituted in equation (57), will reduce 
it to, 
r cos cp = u — f b n. 
C n, being as before perpendicular to B S. 
With the centre C, and radius equal to one third 
the radius of the reflector, describe the circle 
0"V ; with the centre t, the middle of V B, and 
radius t V, describe the circle V O B, which will 
pass through O ; the arc Y O" will equal the arc 
Y O, and the caustic will again be an epicycloid ge- 
nerated by the motion of the circle V O B upon 
O'' V. For, B O is two thirds of B n, and B V two CAUSTICS. 
thirds of B C, and the angle V O B, consequently, 
a right angle ; the angle B" CB=2CB A = 
2CBS = OiV, and C V and V t being equal, the 
arcs O'' V and V O are equal. 
When the radiant passes within the circle, and 
reflection takes place in all directions, there will 
be some position for the incident ray, giving 
u = cos <3°, which will render the denominator in 
equation (57), equal to zero. The caustic in this 
case will be divided into two branches with cusps at 
O", O' and O'" (fig. 37), and infinite branches ex- 
tending along the reflected ray answering to that 
2 u 
value of <p, whose cosine is equal to ; this 
r 
ray will be an asymptote to both branches. 
63. The subject of caustics being one of curiosity 
rather than utility, it is perhaps unnecessary to our 
present purpose to pursue it further ; it will, there- 
fore, be terminated by a general method for finding 
the equation of the caustic curve from that of the 
deviating surface, when the position of the radiant 
is given. 
Let M N (fig. 38), be a section of any deviating 
surface by a principal plane ; Dm a normal at the 
point D ; G D an incident and D II the correspond- 
ing deviated ray. Draw any two rectangular axes 
as A £ and A y. Call x and y, the angles which 
the normal makes with the axes of x and y respec- 
tively, and we shall have CAUSTICS. 
d X 
sin x = cos y = ——, 
d s 
d y 
sin y =cos x = —— ; 
d s 
and calling w and &/ the angles which the incident 
and deviated rays make respectively with the axis 
of x, we get from the triangles G Dm and G' D n, 
sin cp = sin (x — w), 
sin cp' — sin (x — w'), 
or 
d x . d y 
sin cp — cos w — sin 
d s d s 
, d x d y 
sin q>'~ cos co —-—■ — sin co — - • 
d s d s 
these being substituted in the equation, 
give 
sin cp — m sin cp' — 0, 
cos hi.dz — sin co d y — m (cos a)dx — sin w( d y) — 0 . (59) 
Let xy, be the co-ordinates of the point of in- 
cidence D, u (?, those of any point on the incident, 
and tt/(5(, those of any point on the deviated ray; 
since both of these rays pass through the point 84 
CAUSTICS. 
whose co-ordinates are x and y, their respective 
equations will be, 
/? — y — tan co . (a — x) 
P — y — tan co, (« — x) 
from the first we have, 
2 . 2 1 
P—y +«—* t 2 . , 2 1 
—tan m4-1 = sec co = — 
a-x ' cos w 
or 
a — x 
COS co = —11"— -=r ; 
*)* + (!?—-y)s 
and 
# -,2 
1 — cos2 w = smsw = £L 
a . 2 ’ 
a — x + p — y 
or 
P — y 
smw“ 
In the same way we obtain 
a — x 
COS to, “=* -7 -=zr=r CAUSTICS. 
85 
&, — y 
— xYMP—yY 
These values in equation (59) give, 
(a — x)dx— (|9 — y)dy (a — x)dx — {$ — y)dy 
■/ (a _ xy -f- ((9 — yY d(a — xY + (P, — yY 
Now if x be assumed, y becomes known from 
the equation of the curve of intersection of the de- 
viating surface with the principal plane, (which we 
will call the deviating curve), and the point of in- 
cidence is, therefore, determined ; and if the ra- 
diant be also assumed, it being on the incident 
ray, « and p become known, and the incident ray 
will be given in position. These values of « p, x 
and y, and the differentials of the latter, deduced 
from the equation of the deviating curve, being 
substituted in equation (60), will give an equation 
containing only the variables and (?,, which will 
be that of the deviated ray, and may be represen- 
ted by 
F M,) =0 . . . (61) 
If equation (60) be differentiated with reference to 
x and y, the derived equation will be a function of 
« 8, «, Plt x y, and for the same values as before as- 86 
CAUSTICS. 
sumed for the co-ordinates « ft x y, will appertain 
to a second deviated ray, proceeding from a point 
on the deviating curve whose distance from that at 
which the first ray is deviated, is equal to ds. 
This equation, which may be represented by 
f,(«,5,)=0, . . . (62) 
and equation (61), containing but two variables, 
viz : (?7, these co-ordinates become known, and 
determine the point common to two consecutive de- 
viated rays, which is one point in the caustic. 
We have supposed the point of incidence 
to be given, and have found the corresponding 
point of the caustic; if, however, we combine 
equations (61) and (60) with that of the deviating 
curve by eliminating xy, and their differentials, the 
resulting equation will be independent of the co- 
ordinates of the points of incidence, and will be a 
function of « ft «(ft, and may be written, 
F„ (“ P «, P) = 0; 
Now, assuming the position of the radiant, or the 
co-ordinates a and /?, the resulting equation, con- 
taining the variables « and £, will evidently be that 
of the caustic. SURFACES OF ACCURATE CONVERGENCE. 
87 
Surfaces of Accurate Convergence. 
64. We have thus far supposed the deviation to 
take place at spherical surfaces, and have seen that 
for a pencil of any considerable magnitude, the 
rays at a distance from the axis wander or aber- 
rate from the focus into which those immediately 
about the axis are concentrated. It is now pro- 
posed to assume a pencil of any magnitude, and to 
find a deviating surface which shall concentrate all 
its rays accurately to the same focus. 
For this purpose, join the given radiant R, (fig. 
39), and the point A into which the rays are to be 
collected ; take this line as the axis of x, and the 
origin at the focus. Calling u and ul the distances 
from the point of incidence D to any two points 
assumed, one on the incident, the other on the de- 
viated ray, and of which the co-ordinates are « 0, 
and «( ft, we shall have, 
u2 = (« — %y 4- (ft — yY 
u~ = (« — %Y + (ft — y)~, 
— u du — (« — x)dx 4- (i? — y) dy. 
— «,<!*, = (<*,— x) d x 4- (81— y) d y . 
these substituted in equation (60), reduce it to 
du, — mdus=0 . . . (61) 88 
SURFACES OF ACCURATE CONVERGENCE. 
or 
u — 7n uj n —0 . . . (62)' 
Calling c, the distance between the radiant 
and focus, we have, 
u = — x)2 -f- y2 
u = x2 + y~ 
these values substituted in equation (62)', give 
(c—x)2 -\-y2—md x2 -f- y2 + n =0 . . (63) 
which is the equation of a principal section of the 
required surface. It is of the fourth degree. 
If the rays be parallel, (fig. 40), we have 
du — d x, 
substituting this value of d u in equation (61)', 
we get 
d x—m du4— 0 
hence 
x — m u -{- n — 0 
or 
x — m x2-\-y '1 -1-»-=0 . . . (64) SURFACES OF ACCURATE CONVERGENCE. 
89 
Clearing the radical and reducing, we have 
m2 y5 + (m2 — 1) x1—2 nx — na=0, . (65) 
which is an equation of a conic section, the nature 
of which will depend upon m. If m be greater 
than unity, it will be an ellipse ; if equal to unity, 
a parabola ; and less than unity, an hyperbola. 
It is obvious from equation (64), that the origin 
is at one of the foci, since the radius vector which 
is equal to x2 + ?/2, is expressed in rational 
functions of the absciss x. To find at which 
focus the origin is situated, we have, making y — 0, 
„ 2 n , . 
X- — —x 71* — 0 
f»-— 1 
or 
» 
2- r, 
m — 1 
n 
X=Z 
The positive value of x being greater than the 
negative, the origin is at the focus most distant from 
the surface of incidence. 
Adding the values of x, we have the transverse 90 
SURFACES OF ACCURATE CONVERGENCE. 
axis ; taking their difference, we have the distance 
between the foci; the first is, 
2 mn 
ma— f 
which we will call 2 A ; and the second 
2 n 
m‘l — 1 
which will be called 2 E: hence 
a : e :: m: 1 
or, the semi-transverse axis is to the eccentricity, as 
the index of refraction to unity. 
Let MAN, (fig. 41), be the section of an ellip- 
soid of revolution whose semi-transverse axis and 
eccentricity satisfy the above proportion ; parallel 
rays having the direction of the axis A B will be 
brought to the focus F. With F as a centre, and any 
radius less than F A, describe the arc of a circle 
M N ; this will represent the section of a spherical 
surface to which the rays deviated at the ellipsoidal 
surface will be normal, and M A N M, will repre- 
sent a section of an aplanatic meniscus. In this 
case m is greater than unity. If, however, m be SURFACES OF ACCURATE CONVERGENCE. 
91 
less than unity, the deviating surface becomes (fig. 
42), an hyperboloid of revolution whose semi-trans- 
verse axis and eccentricity must satisfy the same 
proportions; the rays being parallel to the axis 
and incident on a plane surface perpendicu- 
lar to that line, we shall have an aplanatic plano- 
convex lens. 
If, n = 0, the general equation (63), will re- 
duce to 
(m2- — 1) y2 + (m2 — 1} x1 +2 cx — c2 — 0, 
which is an equation of a circle, (see Davies’ 
Anal. Geom. Book VII. Art. 26), whose radius is 
c m 
T=r* 
m — I 
and the co-ordinates of whose centre are 
y = o 
Calling d, the distance of the radiant from the 92 
SURFACES OF ACCURATE CONVERGENCE. 
centre of curvature, we have 
y , , c m2 c 
=*=— ; 
~ m2—l m — 1 
hence, 
d: r : : m : 1 
If, therefore, with C as a centre (fig. 43), and 
any radius r, we describe the arc MAN, and the 
distance C O, be taken bearing to r the relation 
given by the above proportion, rays converging to 
the point O, will be deviated by the spherical sur- 
face accurately to some point as F, whose distance 
from C is given by the value of x. With the cen- 
tre F and any radius less than F A, describe the 
arc MSN, and the figure MANS will be a section 
of an aplanatic spherical meniscus. 
If in equation (63), m — — 1, we shall have the 
equation of the curve of accurate convergence by 
reflection, which becomes after reduction, 
4»s jr — (cs— n2)*-4z* (c2 — 7i2)-j-6 cx2(c2-»*)= 0 . (65). 
This is an equation of a conic-section, the nature 
of which will depend upon the value of c2 as com- 
pared with ft2. It will be that of an ellipse if c2< n2, OF THE EYE AND OF VISION. 
93 
or a hyperbola if <r > ir. In case of parallel rays, 
equation (64) becomes, when m =— 1, 
y2 — 2 nx — n2 = 0 
an equation of a parabola. 
Of the Eye and of Vision. 
65. The eye is a collection of refractive media, 
which concentrate the rays of light proceeding 
from every point of an external object, on a tissue of 
delicate nerves, called the retina, there forming an 
image, from which, by some process unknown, 
our perception of the object arises. These media 
are contained in a globular envelope composed of 
four coatings, two of which, very unequal in ex- 
tent, make up the external enclosure of the eye, 
the others serving as lining to the larger of these 
two. 
The shape of the eye is spherical except imme- 
diately in front, where it projects beyond the sphe- 
rical form, as indicated at d e d", (fig. 44), which 
represents a section of the human eye through the 
axis in a horizontal plane. This part is called the 
cornea, and is in shape a segment of an ellipsoid 
of revolution about its transverse axis which coin- 
cides with the axis of the eye, and which has to 
the conjugate axis, the ratio 1,3. It is a strong, 
horny, and delicately transparent coat. 94 
OF THE EYE AND OF VISION. 
Immediately behind the cornea, and in contact 
with it, is the first refractive medium, called the 
aqueous humour, which is found to consist of nearly 
pure water, holding a little muriate of soda and gela- 
tine in solution with a very slight quantity of albu- 
men. Its refractive index is found to be very nearly 
the same as that of water, viz : 1,336, and parallel 
rays having the direction of the axis of the eye 
will, in consequence of the figure of the cornea? 
after deviation at the surface of this humour, con- 
verge accurately to a single point. 
At the posterior surface of the chamber A, in 
contact with the aqueous humour, is the iris, g g, 
which is a circular opaque diaphragm, consisting 
of muscular fibres by whose contraction or expan- 
sion an aperture in the centre, called the jpupil, 
is diminished or increased according to the supply 
of light. The object of the pupil seems to be, to 
moderate the illumination of the image on the re- 
tina. The iris is seen through the cornea, and 
gives the eye its color. 
In a small transparent bag or capsule, immedi- 
ately behind the iris and in contact with it, closing 
up the pupil, and thereby completing the chamber 
of the aqueous, lies the crystalline humour B ; 
it is a double convex lens of unequal curvature, 
that of the anterior surface being least; its density 
towards the centre is found to be greater than at 
the edge, which corrects the spherical aberration 
that would otherwise exist; its mean refractive in- OF THE EYE AND OF VISION. 
95 
dex is 1,384, and it contains a much greater por- 
tion of albumen and. gelatine than the other 
humours. 
The posterior chamber C, of the eye, is filled 
with the vitreous humor, whose composition and 
specific gravity differ but little from the aqueous. 
Its refractive index is 1.339. 
At the final focus for parallel rays deviated by 
these humors, and constituting the posterior sur- 
face of the chamber C, is the retina hh h, which is a 
net work of nerves of exceeding delicacy, all pro- 
ceeding from one great branch O, called the optic 
nerve that enters the eye obliquely on the side of 
the axis towards the nose. The retina lines the 
whole of the chamber C, as far as i i, where the 
capsule of the crystalline commences. 
Just behind the retina is the choroid coat k k, 
covered with a very black velvety pigment, upon 
which the nerves of the retina rest. The office 
of this pigment appears to be to absorb the light 
which enters the eye as soon as it has excited the 
retina, thus preventing internal reflection and con- 
sequent confusion of vision. 
The next and last in order is the sclerotic coat, 
which is a thick, tough envelope d d'd", uniting 
with the cornea at d d" and constituting what is 
called the white of the eye. It is to this coating 
that the muscles are attached which give motion to 
the whole body of the eye. 
From the description of the eye, and what is 
said in article (59), it is obvious that inverted images 96 
OF THE EYE AND OF VISION. 
of external objects are formed on the retina. This 
may easily be seen by removing the posterior coat- 
ing of the eye of any recently killed animal and 
exposing the retina and choroid coating from be- 
hind. The distinctness of these images, and con- 
sequently of our perceptions of the objects from 
which they arise, must depend upon the distance 
of the retina from the crystalline lens. The habit- 
ual position of the retina, in a perfect eye, is near- 
ly at the focus for parallel rays deviated by all the 
humors, because the diameter of the pupil is so 
small compared with the distance of objects at 
which we ordinarily look, that the rays constituting 
each of the pencils employed in the formation of 
the internal images may be regarded as parallel. 
But we see objects distinctly at the distance of a 
few inches, and as the focal length of a system of 
lenses, such as those of the eye, (equation 16'), in- 
creases with the diminution of the distance of the 
radiant or object, it is certain that the eye must 
possess the power of self adjustment, by which 
either the retina may be made to recede from the 
crystalline humor, and the eye lengthened in the 
direction of the axis, or the curvature of the lenses 
themselves altered, so as to give greater conver- 
gency to the rays. The precise mode of this ad- 
justment does not seem to be understood. There 
is a limit, however, with regard to distance, within 
which vision becomes indistinct; this limit is 
usually set down at six inches, though it varies with 
different eyes. The limit here referred to is an OF THE EYE AND OF VISION. 
97 
immediate consequence of the relation between 
the focal distances expressed in equation (16)', for 
when the radiant or object is brought within a few 
inches, the corresponding conjugate or image is 
thrown behind the point to which the retina may 
be brought by the adjusting power of the eye. 
With age the cornea loses a portion of its con- 
vexity, the power of the eye is, in consequence, di- 
minished, and distinct images are no longer formed 
on the retina, the rays tending to a focus behind 
it. Persons possessing such eyes are said to be 
long sighted, because they see objects better at a 
distance ; and this defect is remedied by convex 
glasses, which restore the lost power, and with it, 
distinct vision. 
The opposite defect arising from too great con- 
vexity in the cornea is also very common, particu- 
larly in young persons. The power of the eye 
being too great, the image is formed in the vitre- 
ous humor in front of the retina, and the remedy 
is in the use of concave glasses. Cases are said 
to have occurred, however, in which the prominence 
of the cornea was so great as to render the conve- 
nient application of this remedy impossible, and 
relief was found in the removal of the crystalline 
lens, a process common in cases of cataract, where 
the crystalline loses its transparency and obstructs 
the free passage of light to the retina. 
The fact that inverted images are formed upon 
the retina, and we, nevertheless, see objects erect, 
has given rise to a good deal of discussion. With- 98 
OF THE EYE AND OF VISION. 
out attempting to go behind the retina to as- 
certain what passes there, it is believed that the 
solution of the difficulty is found in this simple 
statement, viz : that we look at the object, not at 
the image. This supposes that every point in an 
image on the retina, produces, without reference 
to its neighboring points, the sensation of the ex- 
istence of the corresponding point in the object, 
the position of which the mind locates some where 
in the axis of the pencil of rays of which this point 
is the vertex ; all the axes cross at the optical cen- 
tre of the eye, which is just within the pupil, and 
although the lowest point of an object will, in con- 
sequence, stimulate by its light the highest point 
of the retina affected, and the highest point of the 
object the lowest of the retina, yet the sensations 
being referred back along the axes, the points will 
appear in their true positions and the object to 
which they belong erect. In short, instead of the 
mind contemplating the relative position of the 
points in the image, the image is the exciting cause 
that brings the mind to the contemplation of the 
points in the object. 
It may be proper to remark here that the base 
of the optic nerve, where it enters the eye, is totally 
insensible to the stimulus of light, and the reason 
assigned for this is, that at this point the nerve is 
not yet divided into those very minute fibres which 
are capable of being affected by this delicate 
agent. OF THE EYE AND OF VISION. 
99 
66. The apparent magnitude of an object, is 
determined, by the extent of retina covered by 
its image. 
If, therefore, R R' (fig. 45), be a section of the 
retina, by a plane through the optical centre C of 
the eye, and AB = l,ab = l, the linear dimensions 
of an object and its image in the same plane, we 
shall have, 
l =c a. —• (66) 
calling s, the distance of the object. And for any 
other object whose linear dimension is 1' and dis- 
tance s/} calling the corresponding dimension of 
the image i. 
, v 
*=ca. — 
' B 
J 
and since C a is constant, or very nearly so, 
' e s’ 
or the apparent linear dimensions of objects are 
as their real dimensions directly, and distances from 
the eye inversely. But may be taken as the 100 
OF THE EYE AND OF VISION. 
measure of the angle B C A = b C a, which is call- 
ed the visual angle, and hence the apparent linear 
magnitudes of objects are said to be directly pro- 
portional to their visual angles. 
Small and large objects may, therefore, be made 
to appear of equal dimensions by a proper adjust- 
ment of their distances from the eye. For exam- 
ple, if x =: tj} we have 
L L 
e e ' 
i 
or 
l'. e. 
*' “ ~T> 
and if l — 1000 feet, e = 20,000 and l' = ,1 of a 
foot, or little more than an inch, 
20,000 x,l 0 , 
e -* —: _ = 2 feet, 
' 1000 
the distance of the small object at which its 
apparent magnitude will be as great as that of 
an object ten thousand times larger, at the dis- 
tance of 20,000 feet. MICROSCOPES AND TELESCOPES, 
101 
Microscopes and Telescopes. 
67. From what has just been said, it would ap- 
pear that there is no limit beyond which an object 
may not be magnified by diminishing its distance 
from the optical centre of the eye. But when an 
object passes within the limit of distinct vision, 
what is gained in its apparent increase of size, is 
lost in the confusion with which it is seen. If, 
however, while the object is too near to be distinctly 
visible, some refractive medium be interposed to 
assist the eye in bending the rays to foci upon its 
retina, distinct vision will be restored, and the mag- 
nifying process may be continued. Such a medi- 
um is called a single microscope, and usually con- 
sists of a convex lens, whose principal focal dis- 
tance is less than the limit of distinct vision, and 
whose index of refraction is greater than unity. 
To illustrate the operation of this instrument, 
let M N (fig. 46), be a section of a double convex 
lens whose optical centre is C; Q,P an object 
in front and at a distance from C equal to the 
principal focal distance of the lens; E the optical 
centre of the eye at any distance behind the lens. 
The rays Q, C and P C, containing the optical 
centre will undergo no deviation, and all the rays 
proceeding from the points Q and P, will be respec- 
tively parallel to these rays after passing the lens ; 
some rays as N E from Q, and M E from P, will 
pass through the optical -centre of the eye, and 102 
MICROSCOPES AND TELESCOPES. 
be the axes of two beams of light whose boun- 
daries will be determined by the pupil, and whose 
foci will be at q and p on the retina, giving the 
visual angle, 
m' en' = p c q; 
or the apparent magnitude of the object P Q, the 
same as if the optical centre of the eye were at 
that of the lens. And this will always be the case 
when an object occupies the principal focus of a 
lens whatever the distance of the eye, provided it 
be within the field of the rays. 
Without the lens, the visual angle is Q, E P < 
P C Q,; hence the apparent magnitude of the ob- 
ject will be increased by the lens. 
Calling x and i, the apparent magnitudes of the 
object as seen with, and without the lens, we shall 
have, 
ru : ’’U 1 : 1 
' CQ Eft CQ EQ 
or 
T =£ —1,(v+7) ' • • • <«> 
by using the notation employed in equation (23), 
and calling E Q, the limit of distinct vision, unity. 
As long as f„ < X, or the principal focal length MICROSCOPES AND TELESCOPES. 
103 
of the lens is less than the limit of distinct vision, 
the apparent size of the object will be increased, 
and the lens may be used as a single microscope. 
We can now understand what is meant by the 
power of a lens or combination of lenses, referred 
to at the close of article (32). —, which was 
F 
// 
there said to measure the power of a lens, we see 
from equation (67), expresses the number of times 
the apparent magnitude of an object is increased be- 
yond that at the limit of distinct vision, by the use of 
the lens ; and whatever has been demonstrated of 
the powers of lenses generally, is true of magnify- 
ing powers. Thus, in equation (22), we have the 
magnifying power of any combination of lenses 
equal to the algebraic sum of the magnifying pow- 
ers taken separate!’/. Should any of the individu- 
als of the combination be concave, they will enter 
with signs contrary to those of the opposite curva- 
ture. 
The power of a single microscope is, equation 
(67), equal to the limit of distinct vision divided 
by its principal focal distance, and its numerical 
value will be greater as its refractive index and 
curvature are greater. 
68. It will be recollected that the last member 
of equation (67), was deduced from equation (18), 
by neglecting the thickness of the lens. Should, 
however, the microscope be an entire sphere, as is 
often the case, the thickness will be equal to twice 104 
MICROSCOPES AND TELESCOPES. 
the radius and ought not to be omitted. By substi- 
tuting 2 r for ts equation (18) reduces to, 
i mr 
_ m\m — 1) — 
1 v f m—\ 
_ — ■ — - 
and supposing the rays parallel, 
1 m[m — 1) m—1 
r (m — 2) r 
or 
r (m —2) 
P" = 2 (m —T)' 
This value of F/; is estimated from the first sur- 
face. When estimated from the optical centre it 
becomes, 
r{m — 2) mr 
P" — 1) T 2(m — 1)’ 
and 
. . (68). 
™ r r 
from which it is obvious that the power will be 
greatest where m is greatest and r least. MICROSCOPES AND TELESCOPES. 
105 
69. To obtain a general expression for the visual 
angle under which the image of an object, placed at 
any distance from a lens, is seen, let Q P (fig.47), 
be an object in front of a convex lens whose 
optical centre is E ; q y its image, and O the posi- 
tion of the eye. Calling the visual angle p O q, 
A, we obtain, by taking the arc equal to its tan- 
gent, the angle being very small, 
A =. JUL — VJP - 
, 0 q o E— E q' 
and representing the distances Q E by/; E q by 
f" ; and E O by d, we have 
J" 
= QP.— 
EO —E q=d—f" 
hence, 
QP f" 
f 'd-r 
Q p 
but —jr is the visual angle, when the eye is at the 
centre of the lens ; calling this A,, we have 
. . (69). 
' Z-1 106 
MICROSCOPES AND TELESCOPES. 
This relation has been obtained on the supposi- 
tion that d and f" are positive on the opposite 
side of the lens from the object; and if, as here- 
tofore, we regard distances estimated in that direc- 
tion negative, which we shall do for sake of uni- 
formity, the equation will remain as at present 
written. 
Now, supposing distinct vision possible for all 
positions of the eye, an examination of equation 
(69) will make it appear, 
1st. That when the object is at a distance from 
the lens greater than that of the principal focus, 
in which case there will be a real image, the lens 
will make no difference in the apparent magnitude 
of the object, provided the eye is situated at a dis- 
tance from it equal to twice that of the image. 
2d. At all positions for the eye between this limit 
and the image, the apparent magnitude of the ob- 
ject is increased by the lens. 
3d. At a position half way between this limit 
and the lens, the apparent magnitude of the object 
would be infinite. 
4th. The eye being placed at a distance greater 
than twice that of the image, the apparent magni- 
tude of the object will be diminished by the lens. 
5th. When the distance of the object from the 
lens is equal to that of the principal focus, in which 
case/" becomes infinite, the apparent magnitude 
will be the same as though the eye were situated 
at the centre of the lens, no matter what its actual 
distance behind the lens. MICROSCOPES AND TELESCOPES. 
107 
6th. In case of a concave lens, f" changing its 
sign, the apparent magnitude of the object will 
always be diminished by the lens. 
The visual angle when the object is placed in 
front of a reflector, (fig. 48), is found in the same 
way. 
q p a P E q 
o q E Q o j ’ 
and representing, as before, E Q, E q, and E O, by 
P Q 
f, f', and d respectively, and the visual angle — 
E Q, 
by A,, we have 
• • • (** 
We shall not stop to discuss this equation. It 
may be remarked, however, that when the reflec- 
tor is convex, the apparent magnitude of the ob- 
ject will be diminished by its use. 
70. We have supposed in the preceding dis- 
cussion, distinct vision to be possible for all posi- 
tions of the eye ; but this we know depends upon 
the state of convergence or divergence of the rays. 
If, however, the image, when one is formed, in- 108 
MICROSCOPES AND TELESCOPES. 
stead of being seen by the naked eye, be viewed 
by the aid of another lens or reflector, so placed 
that the rays composing each pencil proceeding 
from the object shall, after the second devia- 
tion, be parallel, or be within such limits of con- 
vergence or divergence that the eye can command 
them, the object will always be seen distinctly, and 
either larger or smaller than it would appear to the 
unassisted eye, depending upon the magnitude of 
the image, and the power of the lens or reflector 
used to view it. As most eyes see distinctly with 
parallel rays, this second lens or reflector is so 
placed that the image shall occupy its principal 
focus; and where this is the case, we have seen that 
the apparent magnitude of the image will be the 
same as though the eye were at its optical centre. 
Calling the principal focal distance of this lens 
or reflector (f/7) ; d, in equation (69), will be 
f" (f/7), and that equation will become, 
t=<& (7,) 
and if the object P Q, (fig. 49), be so distant that 
the rays composing the small pencil whose base is 
M N, may be regarded as parallel, f" becomes f„, 
and we have, 
i-t (72) MICROSCOPES AND TELESCOPES. 
109 
Equation (71) involves the principles of the 
compound refracting microscope, and refracting 
telescope; and equation (72), which is a particular 
case of (71), relates to the astronomical refracting 
telescope. The lens M N, next the object, is called 
the object or field lens, and m n, the eye lens. The 
magnifying power in the first case, is equal to the 
distance of the image from the field lens divided by 
the principal focal length of the eye lens; and in 
the second, to the principal focal length of the 
field lens, divided by that of the eye lens. 
If instead of a convex, a concave lens be used 
for the eye lens, the combination will be of the 
form used by Galileo, who invented this instrument 
in 1609. In this construction, the eye lens (fig. 50), 
is placed in front of the image at a distance equal 
to that of its principal focus, so that the rays com- 
posing each pencil shall emerge from it parallel. 
The rule for finding the magnifying power of this 
instrument is the same as in the former case ; for 
in equation (69), we have, on account of the prin- 
cipal focal distance of the concave lens being of a 
sign contrary to that of the convex, 
<*=/'-(-00)=/"-HO 
which reduces that equation to 
a ar_ 
w ’ MICROSCOPES AND TELESCOPES. 
or for parallel rays, to 
A F„ 
A/ (O ‘ 
If we divide both numerator and denominator of 
equation (72), by f7/ x (f/7), it becomes, 
1 
A _ CO 
“ 1 ’ 
A 
' F„ 
and calling l the power of the field, and l that of 
the eye lens, we have 
— - — (73) 
A( L 
or the magnifying power of the astronomical tele- 
scope is equal to the quotient arising from dividing 
the power of the eye lens by that of the field lens. 
71. If E (fig. 51), be the optical centre of the 
field, and O that of the eye lens of an astronomi- 
cal telescope, the line E O, passing through the 
points E and O, is called the axis of the instru- 
ment. Let Qf P' be any object whose centre is in 
this axis, and c[ p' its image. Now, in order that 
all points in the object may appear equally bright, MICROSCOPES AND TELESCOPES. 
111 
it is obvious from the figure, that the eye lens must 
be large enough to embrace as many rays from the 
points P' and Q/, as from the intermediate points. 
It is not so in the figure ; a portion, if not all the 
rays from those points will be excluded from the 
eye, and the object, in consequence, appear less 
luminous about the exterior than towards the cen- 
tre, the brightness increasing to a certain bounda- 
ry, within which, all points will appear equally 
bright. The angle subtended at the centre of the 
field lens, by the greatest line that can be drawn 
within this boundary, is called the field of view. 
To find this angle, draw m N and Mwto the oppo- 
site extremes of the lenses, intersecting the image in 
p and q, and the axis in X; then willy?*? be the ex- 
tent of the image of which all the parts will appear 
equally bright. Draw and p E P, the 
angle P E Q, =p E q, is the field of view, which 
will be denoted by $; 
(74) 
but 
m n 
pq = . xr (75) 
1 x o 
to find X O and X r, call the diameter M N of the 
object lens «, that of the eye lens §, and we have 
a : : e x : x o 
« -f|3:^::EX-fxo:xo 112 
MICROSCOPES AND TELESCOPES. 
hence, 
and in the same manner, 
xr =r-. x =/"- . tr-H?jhf-rr+trl ’ 
these values in equation (75), give 
ft/" — « (f J 
Ti r'+(.*,,) ’ 
and this in equation (74), gives, by introducing the 
powers of the lenses, 
8 l — a l 
5=-T+— <76> 
The rays of each of the several pencils emerg- 
ing from the eye lens parallel, will be in condition 
to afford distinct vision, and the extreme rays m O', 
and n O', will be received by an eye whose optical MICROSCOPES AND TELESCOPES. 
113 
centre is situated at O'. If the eye be at a greater 
or less distance than O', from the eye lens, these 
rays will be excluded, and the field of view will be 
contracted by an improper position of the eye. It 
is on this account that the tube containing the eye 
lens of a telescope usually projects a short distance 
behind to indicate the proper position for the eye. 
From the similar triangles pO q and m O' n, we 
have 
0 0’= —.rO = ‘^.(F,) . (77). 
p gr pi— a L ' ' > 
This also applies to the Galilean instrument, 
by changing the sign of l, which will render O O', 
negative. The eye should, therefore, be in front 
of the eye glass in order that it may not, by its 
position, diminish the field of view; but as this is 
impossible, the closer it is placed to the eye glass 
the better. 
When the telescope is directed to objects at dif- 
ferent distances, the position of the image, (equa- 
tion 19), will vary, and the distance between the 
lenses must also be changed. This is accom- 
plished by means of two tubes which move freely 
one within the other, the larger usually supporting 
the object and the smaller the eye lens. 
Through the common astronomical refracting 
telescope objects appear inverted, and through the 114 
MICROSCOPES AND TELESCOPES. 
Galilean erect, as must be obvious on the slightest 
examinations of the figures of these instruments. 
72. The terrestrial telescope is a comipon astro- 
nomical telescope with the addition of what is 
termed an erecting piece, which consists of a tube 
supporting at each end a convex lens. The length 
of this piece should be such as to preserve entire 
the field of view, and its position so adjusted that 
the image formed by the object glass, shall occupy 
the principal focus of the first lens, as indicated in 
figure 52. If the lenses of the erecting piece 
be of the same power, the magnifying power of 
the instrument will be equal to that of an astrono- 
mical telescope having the same object and eye 
lens. 
73. If, now, the object approach the field lens, 
f", in equation (71), will increase and the magni- 
fying power become proportionably greater; but 
this would require the tube containing the eye lens 
to be drawn out to obtain distinct vision, and to 
an extent much beyond the limits of convenience 
if the object were very near. This difficulty is 
avoided by increasing the power of the object lens, 
as is obvious from equation (50) ; and when this 
is carried to the extent required by very near ob- 
jects, the instrument becomes a compound micro- 
scope, which is employed to examine minute ob- 
jects. The compound microscope (fig. 53), not dif- MICROSCOPES AND TELESCOPES. 
115 
faring in principle from the telescope, its magni- 
fying power is given by the equation, 
A _ f" 1 1 
A, (*,) (O yr- 
and substituting for its value in equation 
(2$)"', we have 
A _ 1 l 
, V 'T T; 
A !r) I—,, 
or, writing D for —; and representing, as be- 
fore, the powers of the field and eye lenses by 
L and Z, 
A _ l 
A( D L : 
from which it is obvious that the magnifying power 
may be varied to any extent by properly regu- 
lating the position of the object; but a change in 
the position of the object would require a change in 
the position of the eye glass, and two adjustments 
would, therefore, be necessary, which would be 116 
MICROSCOPES AND TELESCOPES. 
inconvenient. For this reason, it is usual to leave 
the distance between the lenses unaltered and 
to vary only the distance of the object to suit dis- 
tinct vision. It is, however, convenient to have 
the power of changing the distance between the 
glasses, as by that a choice of magnifying powers 
between certain limits may be obtained, and for 
this purpose the object and eye glasses are set in 
different tubes. 
74. If the field lens of the astronomical teles- 
cope be replaced by a field reflector M N, as indi- 
cated in figure 54, we have the common astrono- 
mical reflecting telescope. E being the optical 
centre, d becomes equal to f — (f„), and equa- 
tion (70) reduces to, 
A _Z1 
(O ’ 
and for parallel rays to, 
T-Jr .... (?8). 
hence, the rule for the magnifying power is the 
same as for the refracting telescope. 
Figure (54) represents a reflecting telescope of 
the simplest construction, and it is obvious that the MICROSCOPES AND TELESCOPES. 
117 
head of the observer would intercept the whole of 
the incident light, if the reflector were small, and 
a considerable portion even in the case of a large 
one ; to obviate this, it is usual to turn the axis a 
little obliquely, so that the image may be thrown 
to one side where it may be viewed without any 
appreciable loss of light. By this arrangement, 
the image would, of course, be distorted, but in very 
large instruments, employed to view faint and very 
distant objects, it is not sufficient to cause much if 
any inconvenience. This is Herschel’s instrument. 
75. The obstruction of light is in a great mea- 
sure avoided in the Gregorian telescope, of which 
an idea may be formed from figure 55. 
M N is a concave spherical reflector, having a 
circular aperture in the centre ; an image p q of 
any distant object P Q, is formed by it as be- 
fore ; the rays from the image are received by a 
second concave spherical reflector, much smaller 
than the first, by which a second image p' q', is 
formed in or near the aperture of the first reflec- 
tor and is there viewed through the eye lens m n. 
The distance of the small reflector from the first 
image should be greater than its principal focal 
distance, and so regulated that the second image 
be thrown in front of the eye lens, and in its prin- 
cipal focus. In order to regulate this distance, the 
small reflector is supported by a rod that passes 
through a longitudinal slit in the tube of the in- 
strument, the rod being connected with a screw, as 118 
MICROSCOPES AND TELESCOPES. 
represented in the figure, by means of which a 
motion in the direction of the axis may be com- 
municated to it. 
The apparent magnitudes of the images p q and 
p' q , as seen through the same eye glass at the 
distance of its principal focus, are as their real 
magnitudes ; and the latter are as the distances of 
these images from the centre of the small reflec- 
tor, article (60). But by equation (26), making 
m =— 1, and recollecting that in the case before 
us, c is negative, we have, calling f2, the principal 
focal distance of the second reflector, 
c‘ f2 c 
or 
— = F2 . 
c f2 —c ’ 
hence the magnifying power of the Gregorian teles- 
cope is given by the equation, 
A P F„ 
■ • <»> 
from which it is obvious that the apparent magni- 
tude of the object may be made as great as we 
please by giving a motion to the small reflector dynameter. 
119 
which shall cause its principal focus to approach 
the first image, and drawing out, at the same time, 
the eye lens to keep the rays which enter the eye 
parallel. 
76. If the small reflector be made convex instead 
of concave, we have the modification proposed by 
M. Cassegrain, and called the Cassegrainian teles- 
cope, which is represented in figure 56. Its mag- 
nifying power is given by equation (79). 
77. Sir Isaac Newton substituted for the small 
curved reflector a plane one (fig. 57), inclined 
45° to the axis of the instrument, and so placed as 
to intercept the rays before the image is formed. 
The state of the rays with respect to convergence 
or divergence not being affected by reflection at 
plane surfaces, the image is formed on one side, 
and viewed through the lens supported by a small 
tube inserted in the side of the main tube of the 
telescope. The magnifying power of the Newto- 
nian telescope is given by equation (78). 
Dynameter. 
78. If any telescope, except the Galilean, pro- 
perly adjusted to view distant objects, be directed 
towards the heavens, the field lens may be re- 
garded as an object whose image will be formed 120 
DYNAMETER. 
by the eye lens. The distance of the object in 
this case will be the sum of the principal focal dis- 
tances or (f„ + (f7/)), and this being substituted 
for f in equation (51), we get, by inverting and 
reducing, 
8 F 
r—lfi- (80) 
hence, any linear dimension of the object glass of 
a telescope, divided by the corresponding linear di- 
mension of its image, as formed by the eye glass, 
is equal to the magnifying power of the telescope. 
This is the principle of the Dynameter, a beauti- 
ful little instrument used to measure the magnify- 
ing power of telescopes. 
To understand its construction, let us suppose 
(fig. 58), two circular disks of mother-of-pearl, a 
tenth of an inch in diameter, to be placed one ex- 
actly over the other in the principal focus m of a 
lens E, and with their planes at right angles to its 
axis ; an image of the common centre of the disks 
will be formed on the retina of an eye, viewing 
them through the lens, at m". If one of the disks 
be moved to the position m!, so that its circum- 
ference be tangent to that of the other, the image 
of its centre will be at m!", determined by drawing 
from O, the optical centre of the eye, a line paral- 
lel to that joining the optical centre of the lens and 
the centre of the moveable disk, article (67) ; the DYNAMETER. 
121 
images will, of course, be tangent to each other, 
and the moveable disk will have passed over a dis- 
tance equal to its diameter, viz : one tenth of an 
inch. We now take but one disk, and suppose 
the lens divided into two equal parts by a plane 
passing through its axis; as long as the semi- 
lenses occupy a position wherein they constitute a 
single lens, an image of the pearl will be formed 
as before at m"; but when one of the semi-lenses 
is brought in the position denoted by the dotted 
lines in the figure, having its optical centre at E, in 
a line through m, parallel to ml E, two images, 
tangent to each other, will again be formed ; for, 
all the rays from the centre of the pearl, refracted 
by the semi-lens in this second position, will be 
parallel to m E7, and O ml" is one of these rays. 
It is obvious also, that the distance E E7, through 
which the moveable semi-lens has passed, is equal 
to the diameter of the disk of pearl. 
The dynameter consists of two tubes A B, and 
C D, (fig. 59), moving freely one within the other, 
the larger having a metallic base with an aperture 
in the centre whose diameter is equal to one tenth 
of an inch, over which is placed a thin slip of 
mother-of-pearl P. Tn the opposite end of the 
smaller tube, two semi-lenses E, E,, are made to 
move by each other by means of an arrangement 
indicated in figure 60, wherein n is a right- 
handed screw with, say, fifty threads to an inch ; 
n' is a left-handed screw, with the same number 
of threads, which works in the former about a corn- 122 
DYNAMETER. 
mon axis, and is fastened to the frame that carries 
the semi-lens E. The screw n, is rendered sta- 
tionary as regards longitudinal motion, by a 
shoulder that turns freely within the top of the 
frame ST at r, and works in a nut at v connected 
with a frame that carries the semi-lens E'; this 
screw is provided with a large circular head X Y, 
graduated into one hundred equal parts, which may 
be read by means of an index at X or Y, on the 
frame of the instrument. At t is a spring that 
serves to press the frames against their respective 
screws, to prevent loss of motion when a change 
of direction in turning takes place. 
When the graduated head is turned once round 
to the right, the semi-lens E', is drawn up tjV of an 
inch, while the semi-lens E, is thrust in an oppo- 
site direction through the same distance, making 
in all a separation of the optical centres of A of 
an inch, and the lens is kept symmetrical with re- 
gard to the centre of the instrument. If the screw 
had been turned through but one division on the 
head, the separation would have been tot of ■— 
or TJTo of an inch* 
To u e the instrument, direct the telescope, 
whose power is to be measured, to some distant 
object, as a star, and adjust it to distinct vision; 
turn if off the obje t, and apply the dynameter 
with the pearl end next the eye lens, and an image 
of the object lens will be seen ; turn the graduated 
head, supposed to stand at zero, till two images 
appear and become tangent to each other ; read MICROMETER. 
123 
the number of divisions passed over, and multiply 
it by j~~0, the product will give the diameter of the 
image in inches. Measure by an accurate scale, 
the diameter of the visible portion of the object 
glass, which being divided by the measure of its 
image just found, will give the magnifying power. 
The index will indicate zero, if the dynameter be 
properly adjusted, when the semi-lenses have their 
optical centres coincident. 
This little instrument is the more valuable, be- 
cause it gives, by an easy process, the magnifying 
power of any telescope, having a convex eye lens, 
however complicated. It will not apply to the 
Galilean telescope, because the eye lens is con- 
cave and no image of the object lens is, in conse- 
quence, formed by it. 
Micrometer. 
79. When a telescope is used for certain astro- 
nomical purposes, it is usual to put a number of 
fine wires or spiders’ webs, at the focus of the ob- 
ject glass, to determine when any object, as a star, 
is in the axis of the instrument. These constitute 
what is called a micrometer, which in its simplest 
form is represented in figure 61. A B, is a cir- 
cular diaphragm divided into equal parts by five 
parallel wires, all of which are bisected at right 
angles by a sixth. The diaphragm is so placed in 
the telescope that the point O, being the intersec- 124 
MICROMETER. 
tion of the sixth with the middle one of the five 
parallel wires, shall coincide with its axis. If now, 
the telescope be directed to a body moving through 
the field of view in the direction indicated by the 
sixth wire, the time of its passing each of the 
parallel wires may be noted, and a mean of the 
five observations will give the approximate time of 
the body’s passing the axis of the instrument. 
Another kind of wire micrometer is often used 
with the telescope to measure very small angles. 
It consists of two wires a, c, (fig. 62), which are 
made to move parallel to each other by means of 
fine screws A, C, each screw carrying a fork 
A', C', to which the wires are attached. The 
screws have fifty threads to the inch, and are pro- 
vided with large circular heads graduated into 100 
equal parts each, so that a turn through one division 
on the head, will cause the wire connected with it 
to pass through a distance of of an inch. A 
third wire, perpendicular to the two first, is supported 
by a small diaphragm, disconnected with the screws, 
upon one of the interior edges of which is placed 
a graduated scale in the shape of saw teeth to in- 
dicate the number of entire revolutions of each 
screw, the instrument being so adjusted that the 
index of each head shall mark zero, when the 
wires coincide with each other, and accurately 
bisect a small circular hole in the stationary dia- 
phragm immediately under the middle tooth of the 
scale. 
To ascertain the angular value of one division MICROMETER. 
125 
on the screw head, find by trigonometrical compu- 
tation the angle subtended by any distant and well 
defined object; direct the telescope, with the 
micrometer in its place, upon it, and adjust to dis- 
tinct vision; turn the graduated heads till the ob- 
ject is accurately embraced by the wires, and 
count the number of divisions passed over by each 
head ; add these together and divide the angle re- 
duced to seconds by the sum, the quotient will give 
the value sought. If the object be so near, how- 
ever, that the rays received from it may not be re- 
garded as parallel, a correction will be necessary. 
To view near objects, the eye lens must be drawn 
out, in which case the telescope, equation (71), 
will have an increased magnifying power with a 
corresponding decrease in the value of the micro- 
meter revolution. But the magnifying power 
when the image is in the principal focus, is to that 
when in any other [position, as f., to f" ; equa- 
tions (71) and (72). 
Calling e, the distance of the image from the 
principal focus, we have 
f F P2 
e=f —* p 
" f—*„ f—*n 
and 
f„ :f„ + e: :a:x, 
a, representing the approximate value fonnd by the 
first process, and x the true value. 126 
MICROMETER. 
Example. The length of the object was three 
feet, measured in a direction perpendicular to the 
line of sight; the distance from the object glass 
261,9 yards; the principal focal length of the ob- 
ject glass, 45,75 inches, and the sum of the divi- 
sions passed over by the screw heads 1819. Call 
the angle subtended y. 
R 5yds' 
1st. Tan i~20| gyd,; •< the log. of which is 7-280835, 
and 
y -13". 07". 57 = 787", 57 
hence, 
787,57 
a"T819“°-433- 
f2 1 6493 
2d- 9-., aTOir0-0068' 
then, 
1,2708:1,2770 : :0",433:0",435, the true value 
of one division on the screw head. 
The micrometer is usually provided with seve- 
ral eye lenses, the object of which is to increase 
or diminish the field of view as well as to regulate 
the magnifying power of the telescope. A change THE SEXTANT. 
127 
in the eye lens will not affect the value of the 
micrometer revolution, because the apparent mo- 
tion of the wires will undergo the same change as 
the apparent magnitude of the image. But if the 
object glass be changed, or the micrometer be ap- 
plied to a different telescope with the same eye 
lens, the value of the revolution will be altered, 
and it will be equal to its value in the first teles- 
cope, multiplied into the ratio of the magnifying 
powers of the telescopes, taken inversely. The 
magnifying powers may be easily found by the 
dynameter. 
The Sextant. 
80. This instrument is also employed to mea- 
sure angles, but on a much larger scale. It de- 
pends upon the catoptrical principle explained in 
article (26), and consists essentially of two reflec- 
tors I and H, (fig. 63), which stand at right angles 
to the plane of the instrument, in which is a gradu- 
ated arc A B, of sixty degrees, represented in the 
plane of the paper ; a moveable index and vernier ; 
and the frame work necessary to support these in 
their position, and keep the instrument steady. A 
telescope T, having its optical axis, or line of col- 
limation, as it is ( ailed, parallel to the plai e of the 
graduated arc, and six colored glasses, of different 
shades, three at G, and three at G', are added. 
The colored glasses are susceptible of a motion in 128 
THE SEXTANT. 
their own planes, and at right angles to that of the 
instrument, about hinges at n and nl. The purpose 
of the telescope is to magnify and define the ob- 
jects whose angular distance is to be taken, and 
the colored glasses to qualify their light. 
The reflector I, called the index glass, is at- 
tached to the index arm IV, which is moveable 
about the centre of curvature of the graduated 
arc as a centre, and is made of glass ground per- 
fectly plane with its posterior surface, (that next the 
eye at E), covered with an amalgam of tin and mer- 
cury; the reflector H, called the horizon glass, is also 
plane, having half its anterior surface covered, 
the line separating the covered from the transpa- 
rent half being parallel to the plane of the instru- 
ment, and the latter half lying to the left as indi- 
cated by the position of the eye. The telescope 
is supported by a ring S, attached to a stem, called 
the up and down piece, which admits of a motion, 
by means of a milled screw, perpendicular to the 
line of collimation, the purpose of which motion is 
to render an object seen through the transparent 
part, and another seen by reflection from the 
covered part of the horizon glass, equally bright, 
by bringing the telescope in a position such that 
nearly the same number of rays may be received 
from each. 
Now, a ray of light XT, from the top of a steeple, 
for example, being incident upon the index glass, 
Js reflected to the horizon glass in the direction 
IH, and by the latter to the eye in the direction OF THE ADJUSTMENTS. 
129 
H E, through the telescope, at the same time that 
a ray reaches the eye in the same direction 
through the transparent part of the horizon glass, 
from the point Y ; so that the points X and Y will 
seem to occupy the same position in space. 
X E Y, is the angle subtended at the eye by the 
distance X Y ; but this angle being that made by 
the direct ray X I and the same ray after two re- 
flections, is, article (26), double the angle II M I, 
made by the reflectors. ID being drawn parallel 
to H M, D IM = H MI will be half the angle 
subtended by the object. If, therefore, the angle 
DIF=60°, be divided into 120 equal parts, and 
these be numbered as whole degrees, beginning at 
the line ID, and the zero of the vernier Y be placed 
in the plane of the index glass produced, the reading 
of the instrument will indicate the entire angle 
X E Y. To observe with a sextant, then, it is only 
necessary to hold the plane of the instrument in 
that of the objects and the eye, and cause, by a 
motion of the index arm, the objects apparently to 
coincide, 
Of the Adjustments. 
81. The objects of the principal adjustments are : 
1st, to make the index and object glasses perpen- 
dicular to the plane of the instrument; 2d, to make 
these glasses parallel when the zero of the vernier 
coincides with that of the graduated arc; and 130 
OF THE ADJUSTMENT. 
3d, to make the line of collimation parallel to the 
plane of the instrument. 
To accomplish the first, move the index division 
of the vernier to the middle of the graduated arc, 
or limb, as it is called ; then holding the instru- 
ment horizontal with the index glass towards the 
observer, look obliquely down the index glass so 
as to see the circular arc by direct view and by 
reflection at the same time. If the arc appear 
broken, the position of the glass must be altered 
till it appear continuous, by means of small screws 
that attach the frame of the glass to the instru- 
ment. The horizon glass is known to be perpen- 
dicular to the plane of the instrument when, by a 
sweep of the index, the reflected image of an 
object and the image seen directly, pass accurately 
over each other; and any error is rectified by 
means of an adjusting screw, provided for the pur- 
pose, at the lower part of the frame of the glass. 
The second adjustment is effected by placing 
the index or zero point of the vernier to the zero 
of the limb ; then directing the instrument to some 
distant object, (the smaller the better), if it appear 
double, the horizon glass must, after easing the 
screws that attach it to the instrument, if there be 
no adjusting screw for the purpose, be turned 
around a line in its own plane and perpendicular 
to that of the instrument, till the object appear 
single ; the screws being tightened, the perpendi- 
cular position of the glass must again be ex- 
amined. This adjustment may be rendered miner OF THE ADJUSTMENTS. 
131 
cessary by correcting an observation by what is 
called the index error, which is equal to the angu- 
lar separation of the two images of a single ob- 
ject when the zero of the vernier and that of the 
limb coincide ,* to find its value, move the index 
till the images run into each other and appear as 
one; the arc from zero of the limb to that of 
the vernier will be the index error. This may 
sometimes be measured on the arc O A, over which 
the graduation is continued for that purpose, and 
is said to be measured off the arc, or it may be 
measured on the arc O B, when it is said to be 
measured on the arc. In the first case, it is obvi- 
ous the index error should always be added to the 
observed angle, and in the second subtracted. A 
better way, perhaps, to find the index error, is, to 
turn the instrument on any object, as the sun, for 
example, and cause the images of that body to be 
tangent to each other with the index on the are,' 
then with the index off the arc ; the half differ- 
ence of the readings will be the index error which 
will be positive or negative, according as the latter 
or former reading is the greater. 
Example. 
Reading on the arc — 31.56 ' 
off + 31 .22 
2).34" 
index error — 0.1?" 132 
OF THE ADJUSTMENTS. 
The third adjustment is made by the aid of tw<? 
parallel wires placed in the common focus of the 
telescope for the purpose of directing the observer 
to the centre of the field of view, in which, an ob- 
servation should always be made ; these wires are 
parallel to the plane of the instrument, and divide 
the field of view into three nearly equal parts. 
The sun and moon are made tangent to each 
other, when their angular distance is 90° or more, 
at one of the wires; the position of the sextant is 
then altered so as to bring these bodies to the 
second wire ; if the contact continue, the line of 
collimation is parallel to the plane of the instru- 
ment ; if not, the position of the telescope must 
be altered by means of two adjusting screws con- 
nected with the up and down piece. 
It is important that the index glass, and the 
colored glasses used with it, be perfectly plane; 
for if the faces of each of these be not parallel, 
the observations will be fallacious, the amount of 
error depending upon the angle of the faces and 
the inclination of the incident ray. A sextant in 
which these glasses are defective is not, however, 
entirely useless, because an accurate result may 
be obtained by repeating the observation with the 
glasses revolved in their own planes through an 
angle of 180°, and taking a mean of the two rea- 
dings. The horizon glass, with its shaded glasses, 
should also be plane, though this is perhaps less 
important, because that glass being stationary and 
tlie telescope also, the line of sight and the sur- THE ARTIFICIAL HORIZON. 
133 
faces of this glass preserve the same inclination to 
each other. 
The Artificial Horizon. 
82. To measure directly the altitude of any 
celestial object with the sextant, it would be ne- 
cessary that the object and horizon should both be 
distinctly visible, but this is not always the case, in 
consequence of the irregularity of the ground 
which frequently conceals the horizon from view. 
The observer is, therefore, obliged to have re- 
course to an artificial horizon, which consists 
usually of the reflecting surface of some liquid, 
as that of mercury, contained in a small vessel A, 
(fig. 64), which will arrange itself parallel to the 
natural horizon D A C. A ray of light S A, from 
a star at S, being incident on the mercury 
at A, will be reflected in the direction A E, 
making the angle S A C = C A S', (A S' being 
E A produced), and the star will appear to an eye 
situated at E, as far below the horizon as it is 
actually above it. Now with a sextant whose in- 
dex and horizon glasses are represented at I and 
H, the angle S E S', may be measured ; but 
S E S'= S A S' — A S E, and because A E is ex- 
ceedingly small compared with the distance A S, 
of any celestial object, the angle A S E may be 
neglected, and S E S' will equal S A S', or double 
the altitude of the star ; hence one half the reading 134 
CAMERA LUCIDA. 
of the instrument will give the apparent altitude. 
At sea, the observer has the natural, or sea hori- 
zon as a point of departure, and the altitude may 
be measured directly. 
Camera Lucida. 
83. This little instrument, the invention of Dr. 
Wollaston, is of great assistance in drawing from 
nature. In its simplest form, it consists of a glass 
prism, a section of which is represented by ABCD, 
(fig. 65), with one right angle at A, and the oppo- 
site angle C, 135°. Rays proceeding from a 
point of any object S in front of the face AD, 
enter this face without undergoing any material 
deviation, and being received in succession by the 
faces D C and C B within the limits of total reflec- 
tion, they are reflected, and finally leave the face 
B A, in nearly the same state of divergence as when 
they left the object S. The eye E, being so placed 
that the edge B of the prism shall bisect the pu- 
pil, will receive the rays from the prism and bring 
them to a focus r, on the retina, at the same time 
that it will receive through the half of the pupil 
not covered by the prism, rays proceeding from 
the point P of a pencil placed below on a sheet 
of paper, and bring them to the same focus 
r ; so that the point in the object and point of the 
pencil will appear to coincide on the paper, the 
whole of which will be seen through the uncovered CAMERA OBSCURA. 
135 
half of the pupil, and a picture of the object may 
thus be traced by bringing the pencil in succession 
in apparent contact with its various parts. 
The linear dimensions of the picture will be to 
those of the object, as the distance of the camera 
from the paper, to its distance from the object, 
nearly. 
If the paper be very near, the eye may not have 
the power to bring the rays proceeding from the 
pencil to the same focus with those from the ob- 
ject ; this difficulty is obviated by the use of a con- 
vex lens at L, or a concave one at L/ ; the effect 
of the former being to reduce the divergence of the 
rays from the pencil to the same degree with that of 
those from the object, and of the latter, to increase 
the divergence of the rays from the object, and 
render it the same with that of the rays from the 
pencil. The camera lucida is constructed of vari- 
ous forms, having reference to the facility of using 
it, the optical principle being the same in all. 
Camera Obscura. 
84. This instrument is also used to copy from 
nature, and like the camera lucida, is constructed in 
various ways, one of the best of which, is represent- 
ed by fig. 66. A B C is a 'prismatic lens, which 
is nothing more than a triangular prism with 
one or both of its refracting faces ground to sphe- 
rical surfaces ; it is set in a small box resting on a 136 
CAMERA OBSCURA. 
cylindrical tube t v, that moves freely in a similar 
tube in the top of a dark chamber, formed by up- 
rights or legs, about which, is suspended a cotton 
cloth rendered impervious to light by some opaque 
size. On one face of the box m n, containing 
the prismatic lens, is an opening to admit the light 
from any object in front of the instrument, and on 
one side the cloth has been omitted in the figure 
to show a table X Y, supported by the uprights, 
on which the paper is placed to receive the picture. 
Now, the rays from any point in an object S, will 
enter the face A C of the prismatic lens, be totally 
reflected by A B, and brought by C B, to a focus 
on the paper, from which, owing to the minute ir- 
regularities of its surface, they will be reflected in 
all directions ; and thus a picture of the external 
object S will be painted at S', which may easily 
be traced by a person situated within the folds of 
the cloth forming the dark chamber. The effect of 
the prismatic lens being the same as a convex 
lens, except that the former also changes the direc- 
tion of the axis of a pencil deviated by it, it is ob- 
vious that the surface of the paper should be 
spherical. The image of the object is brought 
accurately to the table XY, by means of the tube 
t v, which admits of a vertical motion in the 
top of the chamber; this tube also admits of a 
horizontal motion, the purpose of which is, to take 
in different objects in succession without changing 
the position of the body of the instrument. THE MAGIC LANTERN. 
The Magic Lantern. 
85. This consists of a small close chamber 
(fig. 67), from one side of which proceeds a tube 
containing usually two convex lenses A and B, with 
an intermediate opening for a glass slide C which 
may be moved freely in a direction at right angles 
to the common axis of the lenses. Within the 
chamber is an Argand lamp D, behind which is a 
concave reflector E. The rays proceeding from 
any point in a figure, painted with some trans- 
parent pigment upon the glass slide and strongly 
illuminated by the lens A upon w'hich the direct 
light from the lamp, as well as that from the reflec- 
tor E is concentrated, will be brought to a focus 
by the lens B, on a screen M N, placed at a dis- 
tance in front of the instrument; here the rays be- 
ing reflected will proceed as from a new radiant, and 
a magnified image of the figure will thus appear 
upon the screen. Should the screen be partially 
transparent, a portion of the rays will be trans- 
mitted, and the image will be visible to an observer 
behind it. 
The linear dimensions of the object or figure, 
will be to those of the image, as their respec- 
tive distances from the lens B ; if, therefore, the 
lens B be mounted in a tube which admits of 
a free motion in that containing the lens A, its 
distance from the figure may be varied at pleasure, 138 
UNEQUAL REFRANGIBILITY 
and the image on the screen made larger or smaller; 
the instrument, at the same time, being so moved as 
to keep the screen in the conjugate corresponding 
to the focus occupied by the glass slide. The in- 
strument with an arrangement by which this can be 
accomplished is called, the phantasmagoria. In 
order, however, that the deception may be com- 
plete, there must be some device to regulate the 
light, so that the illumination of the image may 
be increased with its increase of size, not dimin- 
ished, as it would be without such contrivance. 
Solar Microscope. 
86. This is the same as the magic lantern, ex- 
cept that the light of the sun is used instead of 
lhat from a lamp. D E (fig. 68), is a long reflec- 
tor on the outside of a window shutter, in which 
there is a hole occupied by the tube containing the 
lenses. 
The object to be exhibited is placed near the 
focus of the illuminating lens A, so as to be per- 
fectly enlightened and not burnt, which would be 
ithe case were it at the focus. 
Unequal Refrangibility of Light. 
87. We have hitherto regarded light as it comes 
from the sun or any self-luminous body, as a sim- 
ple principle, and supposed that all its integrant OF LIGHT. 
139 
parts are subject to the same laws of deviation, 
and have the same index. But this is not the 
case ; for, if a beam S S' (fig. 69), of solar light be 
admitted into a dark room through a small hole, 
in a window shutter, for example, and received 
upon a screen XY, it will exhibit a round, luminous 
spot at T, in the direction of S S' produced ; but if 
the face of a refracting prism A B C be interposed, 
the spot T will disappear, and there will be 
formed upon the screen above, an elongated image 
of the sun variously and beautifully colored, be- 
ginning with red on the end in the direction of the 
refracting angle A of the prism, and passing in 
succession through orange, yellow, green, blue, 
indigo, and terminating in violet, making seven in 
all. These colors are not separated by well de- 
fined boundaries, but run imperceptibly into each 
other; nor are the colored spaces of the same 
length. The following table exhibits the relative 
lengths of these spaces as obtained by Sir Isaac 
Newton with the glass prism used by him, and by 
Fraunhofer, with a prism made of flint glass. 
Newton. 
Fraunhofer. 
Red 
45 
56 
Orange 
27 
27 
Yellow 
47 
27 
Green 
60 
46 
Blue 
60 
48 
Indigo 
48 
47 
Violet 
80 
100 
Total length 
360 
360 140 
UNEQUAL REFRANGIBILITY 
This colored image, called the solar spectrum, 
is accounted for, by supposing white light to be com- 
posed of an almost infinite variety of elements dif- 
fering from each other in the degree with which 
they are deviated by refraction, and that these ele- 
ments are divided into seven classes distinguished 
by the colors of the spectrum, the elements of each 
class also differing from each other, within certain 
limits, in the amount of their deviation. 
From equation (8), we have, 
m_iin(a±£)_ 
sm a 
in whice <3f, is the refracting angle of the 
and s the angle of deviation ; and from which, 
because a is constant, we may obtain, 
im = dS 
sm a 
or 
dS~ dm. . . .(82). 
cos (a -j- o) 
From this equation it is obvious, that as m 
varies, <5 also varies, and that we have only to 
attribute to m, different values within certain 
limits, beginning with the red rays for which it is OF LIGHT. 
141 
least, and terminating with the violet for which it 
is greatest, to obtain the deviation necessary to 
embrace the entire spectrum. 
Let A be a refracting prism (fig. 70), made of 
any transparent medium, with its edges placed 
horizontally ; mn a graduated circle, to the centre 
of which a small telescope is attached in such a 
manner that its line of collimation shall move in 
a plane parallel to that of the graduated circle, 
which is held in a position at right angles to the 
edges of the prism. The telescope, being pro- 
vided at a solar focus with a fine wire perpendi- 
cular to the plane of the circle, is directed to some 
distant luminous object, and the reading of the 
vernier noted. It is then directed so as to re- 
ceive the colored rays from the prism, and the read- 
ing again noted when the prism is turned to the 
position giving the deviation a minimum. We 
shall then have 
R0Cai = DCS -f-DSC 
or neglecting the very small angle subtended by 
D C at the distance of the object, 
rT =« d C S, 
which is the difference of the readings ; and this 
in equation (81), will give the value of m. 142 
DISPERSION OF LIGHT. 
If the color occupying the middle of the spec- 
trum be taken, we shall find the value of m 
which answers to what is called the mean deviation, 
and which is the same as that given in the table 
of article (19). 
This property of white light, by which its seve- 
ral elements give different indices of refraction with 
the same medium, is called its unequal refran- 
gibility. 
If a hole be made in the screen (fig. 69), at any 
one of the colors, as green, for example, and this 
color, after passing, through, be deviated by a 
second prism F, no further decomposition will be 
found to take place, but a green image, of the 
shape and size of the hole in the first screen, will 
be formed upon a second screen held behind at G'; 
and this being true of every part of the spectrum, 
each of the seven colors, is said to be homoge- 
neous or primary. 
The colors of the spectrum being received, each 
upon a separate mirror, (fig. 71), may, by varying 
the relative position of the mirrors, be reunited, by 
reflection, on a screen at W when a white spot 
will be formed as though it were illuminated with 
common light. 
Dispersion of Light. 
88. From what has already been said, it is ob- 
vious, that white light may be decomposed,analyzed, DISPERSION OF LIGHT. 
143 
or separated into its elementary colors, by re- 
fraction. The act of such separation is called, 
the dispersion of light, and that property of any 
medium by which this is performed is called, its 
dispersive power. 
Supposing the incident beam perpendicular to 
the first face of the prism, the angle of incidence 
on the second will be equal to its refracting angle 
(23) ; and calling y the angle of emergence, there 
will be, because of the relation 
sin ip — m sin a, (83) 
«5 = i// — a, 
an increase in 6, for every increase of a from zero 
to the limit of total reflection. It is evident, there- 
fore, that, by supposing m to vary, two prisms may 
be made of different media, whose refracting an- 
gles shall be so related as to give the same mean 
refraction ; but when this is the case, it is rarely 
found that the lengths of the spectra will be equal, 
the red rays being more, and the violet less refrac- 
ted in one than in the other. The dispersive power 
of the medium of which the prism giving the 
longer spectrum is made, will, therefore, be the 
greater. If the refracting angle of this prism be 
diminished, the length of its spectrum will be di- 
minished as well as its mean deviation, so that, the 
lengths of the spectra may become equal, when the 144 
DISPERSION OF LIGHT. 
mean deviations are very unequal. It is, hence 
usual to take as a measure of the dispersive 
power of any medium, the angle R n V, (fig. 72), 
subtended by the spectrum, divided by the angle 
T n G, of mean deviation. From equation (83) 
we have 
, sin ip — sin a 
m — 1 — . , 
sin a 
and if the angles y, and a be small, 
_i xp~a • 
a 
denoting by tnv, mr and m, the indices corres- 
ponding to the extreme violet, extreme red and 
middle rays respectively, we shall have, 
. I 
*nv — 1 = T n v —, 
m~ — 1 *= t n R —, 
a 
I 
m — Utao —, 
a 
and calling D, the dispersive power, we obtain, 
according to the rule just given, 
(mv— 1)--(mr— 1) ,Q/1, 
_ - = — . . (84). 
m — I in — l DISPERSION OF LIGHT. 
145 
In this way the dispersive powers of the sub- 
stances named in the following table, as well as 
those of a great many others, have been obtained, 
having previously found the indices of refraction 
for the extreme and middle rays of the spectrum 
formed by each substance. 
Table of Dispersive Powers. 
Substances. 
mv — mr 
mv — mr 
m — 1 
Realgar melted, 
0.267 
0.394 
Chromate of Lead, 
0.262 
0.388 
Oil of Cassia, 
0.139 
0.089 
Flint Glass, 
0.050 
0.032 
Crown Glass, 
0.033 
0.018 
Olive Oil, 
0.038 
0.018 
Water, 
0.035 
0.012 
Muriatic Acid. 
0.043 
0.016 
89. There is a circumstance connected with 
this subject which should be carefully noticed, 
owing to its importance in the construction of 
lenses. If the lengths of spectra formed by two 
prisms of different media be the same, the colored 
spaces in the one will not, in general, be equal in 
length to the corresponding spaces of the other. 
This circumstance has been called the irration- 
ality of dispersion. 146 
CHROMATIC ABERRATION. 
Chromatic Aberration. 
90. It follows from the unequal refrangibility of 
the elements of white light, that the action of a 
lens, (fig. 73), will be, to separate these elements 
and direct them to different foci, since the value 
of f", in equation (19), depends upon that of m. 
1 Xl 
Substituting in that equation — - for — + 
in the case of a double convex lens ; and writing 
fv, and fr for the focal distances of the violet 
and red rays, we obtain 
Jr=_K_„.l+.L 
in which mv, being greater than mr, fv, will be 
less and the violet rays will be brought to 
a focus soonest. This deviation from accurate 
convergence, caused by the unequal refrangibility 
of the elements of white light, when deviated by a 
lens, is called chromatic aberration, and depends 
upon the nature of the lens and not on its figure. 
It is measured, along the axis of the lens, by the 
value of fr—fv. CHROMATIC ABERRATION. 
147 
The intersection of the cone of violet rays, 
with that of the red rays, will give what 
is called, the circle of least chromatic aberra- 
tion. The diameter and position of this circle 
can readily be found. From the point s, (fig. 73), 
demit the perpendicular sO =y to the axis ; this 
will divide fr —fv, into two parts Y O = x, and 
O r = w; and calling the semi-aperture of the 
lens a, we shall obtain from the similar triangles 
of the figure, 
y w x 
whence we deduce 
wJrx —fr fv “ ~ (fr +./*) 
or 
fr \fo /qe\ 
y ‘TT+yr (85) 
The denominator of this expression is equal to 
twice the mean value of f", £nd therefore, 
2y= «(/.-/,) .jr„ 148 
CHROMATIC ABERRATION. 
and from equation (19), we have 
1 1 mv — 1 mr — 1 mv — mr 
fv f7 “ 9 9 V~ 
or 
e. r mv mr 
Jt Jv V 
by substituting /,/2, for f .f, to which it is 
nearly equal. 
Substituting the value of ?, from second equa- 
tion of group (A), the above becomes 
f f - mv—mr f"% 
»—1 ’ • 
hence, 
mv — mr f f /CcN 
2y = o._ — d. . . (86) 
m — l r 
In the case of parallel rays, the last factor is 
unity, from which we conclude, that the diameter 
of the least circle of chromatic aberration is equal 
to the semi-aperture of the lens, multiplied by the 
dispersive power. ACHROMATISM. 
149 
The distance of this circle from the lens is, 
fo + X —fo + a ~ • 
V 
replacing by its value in equation (85), we have 
CL 
f I x — fr /cm 
■/'+x—jn-7r- ■ ■ ■ m 
The effect of chromatic aberration is to give 
color to the image of an object, and to produce 
confusion of vision in consequence of the different 
degrees of convergence in the differently colored 
rays proceeding from the same points of an object. 
The vertices of the cones composed of these rays, 
lying in the axis, every section perpendicular to this 
line will have its brightest point in the centre, and 
the yellow rays converging nearly to the mean focus, 
and having by far the greatest illuminating pro- 
perty, the bad effects which would otherwise arise 
from this aberration are in part destroyed. Be- 
sides, these effects may be lessened by reducing 
the aperture of the lens, though not in the same 
degree as those arising from spherical aberration. 
Achromatism. 
91. It is, then, impossible, by the use of a sin- 150 
ACHROMATISM. 
gle homogeneous lens, to deviate a beam or pencil 
of white light accurately to a single focus, and, 
consequently, impossible, by the use of such a 
lens, to form a colorless image of any object; 
both, however, may be done by the union of two 
or more lenses of different dispersive powers. 
The principle according to which this may be ac- 
complished, is termed Achromatism, and the com- 
bination is said to be achromatic. 
Let us suppose two lenses of different disper- 
sive powers placed close together, the power of 
the combination will, equation (22), for any one of 
the elementary colors as red, be 
1 mr — 1 mr‘ — 1 
Fr Q Q 
and for violet, 
1 mv — 1 mv> — 1 
F» 9 ' 91 
If Fr and Fe, were equal, the chromatic aber- 
ration, as regards these colors, would be destroyed; 
equating them we have, 
(»r—i)e' + («r'—!) 9 = (**„—l) ?' + (»v—i)e ACHROMATISM. 
151 
whence, 
q (mv — 1) — (mr — l) mv — mr 
q1 (mr,— 1) — (wv — 1) mv< — mr<' 
the second member being negative because mv, 
is greater than mr. 
Multiplying both members of this equation by 
——it may be put under the form, 
m— 1 
mi — 1 mv — mr 
jz... (88) 
m -— 1 mv> — mr< 
Q m — 1 
The second member expresses the ratio of the 
dispersive powers of the media, and the first the 
inverse ratio of the powers of the lenses for the 
mean rays ; this being negative, one of the lenses 
must be concave the other convex, and the powers 
of the lenses being inversely as their focal dis- 
tances, we conclude, that chromatic aberration, as 
regards red and violet, may be destroyed by uni- 
ting a concave with a convex lens, the principal 
focal lengths being taken in the ratio of their dis- 
persive powers. 
The usual practice is to unite a convex lens of 
crown glass with a concave lens of flint-glass, the 
focal distance of the first being to that of the 152 
ACHROMATISM. 
second as 33 to 50, these numbers expressing the 
relative dispersive powers as determined by ex- 
periment. The convex lens should have the 
greatest power, and, therefore, be constructed 
of the crown-glass ; otherwise, the effect of 
the combination would be the same as that of 
a concave lens with which it is impossible to form 
a real image. 
To illustrate, let parallel rays be received by the 
lens A (fig. 74) ; its action alone would be, to 
spread the different colors over the space V R, 
whose central point m is distant from A, 33 units 
of measure, (say inches), the violet being at V and 
red at R ; the action of the lens B alone would be, 
to disperse the rays as though they proceeded 
from different points of the line R', equal to 
Y R, whose central point m', is distant from 
B = 50 inches, the violet appearing to proceed 
from V' and red from R'; and the effect of the 
united action will be, to concentrate the red and 
R 
violet at ~y~> whose distance from the lens is 
equal to the value of F, deduced from the formula 
111 l . , 
V =33-50“ 906 lnchcs' 
or 
f = 97.06 inches. 
Now if any one of the colors, orange for ex- ACHROMATISM. 
153 
ample, at O, in the space R V, were thrown by the 
convex lens just as far in advance of the centre m, 
as the same color at O' in the space V' R', is 
thrown by the concave lens behind the centre m 
it is obvious that this color would also be united 
R 
with the violet and red at ~y~f by the joint action 
of both lenses ; and the same would be true of any 
other color. But owing to the irrationality of 
dispersion of the media of which these lenses are 
composed, no such union can take place, the 
mean value of in! O', being greater than that of 
m O ; hence this color will not be united with the 
red and violet, and the distance from the point 
-y~} at which it will be thrown, will be equal to 
(ra'0,— ra0), laid off towards the lenses. The 
same being true of the remaining colors, except 
as regards the distance at which they are found, 
some being to the right, others to the left of 
it follows, that an image formed by such a 
combination of lenses will be fringed with color; 
and this is found to be the case in practice, the 
colors of the fringe constituting what is called a 
secondary spectrum. An additional Jens is some- 
times introduced to complete the achromaticity of 
this arrangement. 
92. If two lenses, constructed of media between 154 
ACHROMATISM. 
which there is no irrationality of dispersion, be 
united according to the conditions of equation 
(88), the combination would be perfectly achroma- 
tic. It is found that between a certain mixture of 
muriate of antimony with muriatic acid, and 
crown-glass, and between crown-glass and mer- 
cury in a solution of sal ammoniac, there is little 
or no irrationality of dispersion. These sub- 
stances have therefore been used in the construc- 
tion of compound lenses which are perfectly 
achromatic. Figure (74)' represents a section of 
one of these, consisting of two double convex 
lenses of crown-glass, holding between them, by 
means of a glass cylinder, a solution of the muri- 
ate in the shape of a double concave lens, the 
whole combined agreeably to the relations ex- 
pressed by equation (88). The focal distance 
of the convex lenses is determined from equa- 
tion (22). 
93. From equation (86) we infer, that the cir- 
cle of least chromatic aberration is independent of 
the focal length of the lens, and will be constant, 
provided, the aperture be not changed. Now, by 
increasing the focal length of the object glass of any 
telescope, the eye lens remaining the same, the 
image is magnified ; it follows, therefore, that by 
increasing the focal length of the field lens, we may 
obtain an image so much enlarged that the color 
will almost disappear in comparison. Besides an in- 
crease of focal length, is attended with a diminu- ABSORPTION OF LIGHT. 
155 
tion of the spherical aberration. This explains 
why, when single lenses only were used as field 
glasses, they were of such enormous focal length, 
some of them being as much as a hundred to a 
hundred and fifty feet. The use of achromatic 
combinations has rendered such lengths unneces- 
sary, and reduced to convenient limits, instruments 
of much greater power than any formerly made 
with single lenses. 
Absorption of Light. 
94. If a beam of white light be received upon 
any medium of moderate thickness, it will, in 
general, be divided into three parts, one of which 
will be reflected, another transmitted, and the 
third lost within the medium, or as it is termed, 
absorbed. The quantity absorbed is found to 
vary not only from one medium to another, but 
also in the same medium for the different colors ; 
this will appear by viewing the prismatic spectrum 
through a plate of almost any transparent colored 
medium, such as a piece of smalt blue glass, 
when the relative intensity of the colors will ap- 
pear altered, some colors being almost wholly 
transmitted, while others will disappear or become 
very faint. Each color may, therefore, be said to 
have, with respect to every medium, its peculiar 
index of transparency as well as of refraction. 156 
ABSORPTION OF LIGHT. 
The quantity of each color transmitted, is found 
to depend, in a remarkable degree, upon the 
thickness of the medium, for, if the glass just 
referred to be extremely thin, all the colors are 
seen; but if the thickness be about of an inch, 
the spectrum will appear in detached portions, 
separated by broad and perfectly black intervals, 
the rays corresponding to these intervals being 
totally absorbed. If the thickness be diminished, 
the dark spaces will be partially illuminated ; but 
if the thickness be increased, all the colors be- 
tween the extreme red and violet will disappear. 
Sir John F. W. Herschel conceived that the 
simplest hypothesis with regard to the extinction 
of a beam of homogeneous light, passing through 
a homogeneous medium is, that for every equal 
thickness of the medium traversed, an equal 
aliquot part of the number of rays which up to 
that time had escaped absorption, is extinguished. 
71 
That is, if the th part of the whole number of 
m 
rays, which will be called c, of any homogeneous 
beam which enters a medium, be absorbed on 
passing through a thickness unity, there will re- 
main, 
n m — n 
m m ’ 
71 
and if the a part of this remainder be ab 
m ABSORPTION OF LIGHT. 
157 
sorbed in passing through the next unit of thick- 
ness, there will remain 
m — n n (m — n) m—n 
c > —L. c c, 
m m4 m3 
and through the third unit 
2 , , 
m — n n[m — n) / m — n\i 
C — c = ( ) c, 
m2, m3 V m y 
and through the whole thickness denoted by t units, 
(m — 1 n sm — »\i_1 ✓ m — n\l 
my m\m J \ m J 
So that, calling c the number of equally illumina- 
ting rays of the extreme red in a beam of white 
light, c' that of the next degree of refrangibility, 
c" that of the next, and so on, the beam of white 
light will, according to Sir J. H., be represented by 
c + c'4 c" + c" -f- &c. 
and the transmitted beam after traversing a thick- 158 
ABSORPTION OF LIGHT. 
ness t, by 
cyl -(- c' yl -j- c"y"1 -f- &c. . . . (89) 
7TI 71 
Wherein y represents the fraction —, which 
will depend upon the ray and the medium, and 
will, of course, vary from one term to another. 
From this it is obvious, that total extinction will 
be impossible for any medium of finite thickness ; 
but if the fraction y be small, then a moderate 
thickness, which enters as an exponent, will reduce 
the fraction to a value perfectly insensible. 
Numerical values of the fractions y, y', y", &c., 
may be called the indices of transparency of the 
different rays for the medium in question. 
There is no body in nature perfectly transpa- 
rent, though all are more or less so. Gold, one of 
the densest of metals, may be beaten out so thin 
as to admit the passage of light through it: the 
most opaque of bodies, charcoal, becomes one of 
the most beautifully transparent under a different 
state of aggregation, as in the diamond, “ and all 
colored bodies, however deep their hues and how- 
ever seemingly opaque, must necessarily be ren- 
dered visible by rays which have entered their sur- 
face ; for if reflected at their surfaces, they would 
all appear white alike. Were the colors of bodies 
strictly superficial, no variation in their thickness 
could effect their hues ; but so far is this from be- ABSORPTION OF LIGHT. 
159 
ing the case, that all colored bodies, however in- 
tense their tint, become paler by diminution of 
thickness. Thus, the powders of all colored 
bodies, or the streak they leave when rubbed on 
substances harder than themselves, have much 
paler colors than the same bodies in mass.” 
95. By viewing the prismatic spectrum through 
media possessing different absorptive powers for 
the different rays, Sir David Brewster has been 
able to detect red rays in the blue and indigo 
spaces ; yellow in the red and blue, and blue in the 
red. He has, moreover, been able to obtain white 
light from almost every part of the spectrum by 
absorbing the excess of those colors which predo- 
minate, and he hence infers that the solar spec- 
trum consists of three separate spectra of red, yel- 
low and blue, all of equal length and occupying 
the same space; the red having its maximum in- 
tensity about the middle of the red space, the maxi- 
mum of the yellow being about the middle of the 
yellow, and that of the blue, between the blue and 
indigo. The remaining colors of the spectrum, 
viz : orange, green, indigo and violet, he regards 
as resulting from the superposition of these three. 
Thus, let A C (fig. 75), represent the spectrum ; 
the ordinates of the curve ARC, the number of 
red rays at the corresponding points of the spec- 
trum ; the ordinates of the curve A Y C, the same 
for the yellow; and those of the curve ABC, the 
same for the blue. Now, at every point of the 160 
ABSORPTION OF LIGHT. 
spectrum there will be three ordinates, one of red, 
one of yellow, and one of blue ; and if one of these 
be selected so that portions may be laid off on the 
others bearing to this the relations which exist 
among the numbers expressive of the quantity of 
each of the three colors necessary to form white 
light, the remaining portions of these latter ordi- 
nates will express the excess of those colors which 
predominate. If these be blue and yellow, for in- 
stance, they will mark the green space in the spec- 
trum, this latter color being known to result from 
the mixture of the former ; if red and yellow, the 
orange space ; if red and blue, the indigo or violet, 
according to the proportions. 
Under this view of the constitution of the solar 
spectrum, red, yellow and blue are called primary 
colors, each possessing a refractive index varying 
in numerical value, between those corresponding 
to the extremes of the spectrum. Whenever, 
therefore, the index of refraction of any particular 
color is referred to, it must be understood as rela- 
ting to that part of the spectrum marked by the 
middle of this color, and will belong alike to each 
of the three primary colors from whose union both 
white light and the particular color result. 
96. When the spectrum is formed from light 
proceeding through a narrow slit, say about ~ of 
an inch broad, the refracting edge of the prism 
being parallel to the length of the slit, it is found, 
on examination through a telescope, to be crossed INTERNAL REFLECTION. 
161 
at right angles to its length, or parallel to the edge 
of the prism, by a series of dark parallel lines from 
one end to the other. They are about 600 in 
number, varying in distinctness, the largest sub- 
tending at the distance of the spectrum from the 
prism an angle of from 5" to 10", and the 
distances between them differing from each 
other. These lines are found in the spectra pro- 
duced by all solid and liquid bodies, and whatever 
be the lengths of the spectra or colored spaces, 
they always occupy the same relative position 
within these spaces, provided the light coming 
either directly or indirectly from the sun be used. 
Similar lines are observed when the light of the 
fixed stars is employed, but they have been found 
to vary both in position and intensity. 
The boundaries of the colored spaces of the 
spectrum being but ill defined, these fixed lines 
afford the means, which without them would be 
wanting, to determine with accuracy the refractive 
and dispersive powers of bodies. 
Internal Reflection. 
97. When an object is seen by reflection from 
a plate of glass, the faces of which are not paral- 
lel, it usually appears double. This is owing to 
the reflection which takes place at the second as 
well as first surface, and the image from the former 162 
INTERNAL REFLECTION. 
will be brighter as the obliquity or angle of inci- 
dence of the incident rays becomes greater. In 
this we have supposed the surrounding medium to 
be the atmosphere, between which and glass there 
is a great difference in refractive powers ; but if 
the second surface of the glass be placed in con- 
tact with water, the brightness of the image from 
that surface will be diminished ; if olive oil be 
substituted for the water, the diminution will be 
greater, and if the oil be replaced by pitch, softened 
by heat to produce accurate contact, the image 
will disappear. If, now, the contact be made with 
oil of cassia, the image will be restored ; if with 
sulphur, the image will be brighter than with oil 
of cassia, and if with mercury or an amalgam, as 
in the common looking-glass, still brighter, much 
more so indeed than the image from the first 
surface. 
The mean refractive indices of these sub- 
stances are as follows : 
Air, 
1.0002 
Water, 
1.336 
Olive Oil, 
1.470 
Pitch, 
1.531 to 1.586 
Plate glass, 
1.514 to 1.583 
Oil of Cassia, 
1.641 
Sulphur, 
2.148 
Taking the differences between the index of re- 
fraction for plate glass and those for the other 
substances of the table, and comparing these dif^ THE RAINBOW. 
163 
ferences with the foregoing statement, we are 
made acquainted with the fact, which is found to 
be general, viz : that when two media are in per- 
fect contact, the intensity of the light reflected at 
their common surface will be less, the nearer 
their refractive indices approach to equality; and 
when these are exactly equal, reflection will cease 
altogether. 
98. Different substances, we have seen, have in 
general, different dispersive powers. Two media 
may, therefore, be placed in contact for each of 
which the same color as red, for example, may 
have the same index of refraction, while for the 
other elements of white light, the indices may be 
different; when this is the case, according to what 
has just been said, the reel would be wholly transmit- 
ted, while portions of the other colors would be 
reflected and impart to the image from the second 
surface the hue of the reflected beam ; and this 
would always occur, unless the media in contact 
possessed the same refractive and dispersive 
powers. 
The Rainboiv. 
99. The rainbow is a circular arch, frequently 
seen in the heavens during a shower of rain, in a 164 
THE RAINBOW. 
direction from the observer opposite to that of the 
sun. 
If A B C (fig. 76), be a section of a prism of 
water at right angles to its length by a vertical 
plane, and Sr a beam of light proceeding from 
the sun ; a part of the latter will be refracted at r, 
reflected at D, and again refracted at r', where the 
constituent elements of white light, which had 
been separated at r, will be made further diver- 
gent, the red taking the direction r' R, and the 
violet the direction r' V making, because of its 
greater refractive index, a greater angle than the 
red with the normal to the refracting surface at r'. 
To an observer whose eye is situated at E, the 
point r' will appear red, the other colors passing 
above the eye; and if the prism be depressed so 
as to occupy the position A' B' C', making r" Y't 
parallel to r' V, the point r" would appear of a 
violet hue, the remaining colors from this position 
of the prism falling below the eye. In passing 
from the first to the second position, the prism 
would, therefore, present, in succession, all the 
colors of the solar spectrum. If now the faces of 
the prism be regarded as tangent planes to a sphe- 
rical drop of water at the points where the two 
refractions and intermediate reflection take place, 
the prism may be abandoned and a drop of water 
substituted without altering the effect; and a 
number of these drops existing at the same time 
in the successive positions occupied by the prism THE RAINBOW. 
165 
in its descent, would exhibit a series of colors in 
the order of the spectrum with the red at the top. 
A line ES passing through the eye and the sun, is 
always parallel to the incident rays; and if the 
vertical plane revolve about this line, the drops will 
describe concentric circles, in crossing which, the 
rain in its descent will exhibit all the colors in the 
form of concentric arches having a common cen- 
tre on the line joining the eye and the sun, pro- 
duced in the front of the observer. When this 
line passes below the horizon, which will always 
be the case when the sun is above it, the bow will 
be less than a semi-circle ; when it is in the hori- 
zon, the bow will be semi-circular. 
To find the angle subtended at the eye by the 
radii of these colored arches, let A B D (fig. 77), 
be a section of a drop of rain through its centre ; 
S A the incident, A D the refracted, D B the re- 
flected, and B R the emergent rays. Call the 
angle C A m = the angle of incidence, 9, and the 
angle C A D = the angle of refraction, 9'; the 
angles subtended by the equal chords A D and 
D B, x; and the angle A C B, 6. Then we 
we shall have 
6 = 2?r — 2x\ 
and if there be two internal reflections (fig. 78), 
there will be three equal chords, in which case, 
0 = 2 n — 3 x; 166 
THE RAINBOW. 
and generally, for n internal reflections, 
6 =2 n — n -{- 1 • * . . . . (90) 
but in each of the triangles whose bases are the 
equal chords, and common vertex the centre of the 
drop, 
— 2<jp' 
and this in equation (90) gives, on reduction, 
d = 2(n+ 1) <p' — (n — \)n . (91). 
Because the chords are all equal, the last angle 
of incidence C B D within the drop in fig. (77), 
or OBD' in fig. (78), is equal to the angle of 
refraction CAD, and hence the angle of emer- 
gence C B m' is equal to the angle of incidence 
C A m. 
The angle A O B in fig. (77), is the supple- 
ment of the total deviation of the emergent from 
the incident ray, and is equal to the angle B E F 
subtended by the radius of the bow ; in fig. (78), 
it is the excess of total deviation above 180°. THE RAINBOW. 
167 
Calling this angle 8, we shall have 
8 =. T (2 q> — 6); 
the upper sign referring to fig. (77), and the lower 
to fig. (78) ; replacing d by its value in equation 
(91), the above reduces to 
d = T(2(p — 2(» + 1)<p' + »-1 • n) • .(92) 
this, with equation 
sin cp = m . sin cp', (93) 
will enable us to determine the value of 8, when q> 
and m are given for any particular color. 
For any value of <r assumed arbitrarily, 8 will, 
in general, correspond to rays of the same color 
so much diffused as to produce little or no im- 
pression upon the eye ; but if <3° be taken such as 
to give 5 a maximum or minimum, then will the 
rays of the color, corresponding to m, emerge paral- 
lel, or nearly so, for a small variation in the angle 
g> on either side of that from which this maxi- 
mum or minimum value of d results; hence, the 168 
THE RAINBOW. 
rays which enter the eye in this case will be suffi- 
ciently copious to produce the impression of color, 
and these are the rays that appertain to the rain- 
bow. To find this value of 9>, we have, from 
equation (92), 
d 8 d ® 
but from equation (93) we obtain 
d <jp' cos <p 
dtp m cos <p' 
hence, 
= 2(» + 1) . . 94) 
dtp m cos <p 
or 
1 COS <p 
n -f-1 m cos (p 
Clearing the fraction, squaring both members, 
and adding 
ein3 cp = m,* sin3 g> THE RAINBOW. 
169 
and reducing, we get 
cos q> = ...... (95) 
»2 
For one internal reflection, which answers to 
figure (77), 
tos <p=\/EEl; 
3 
and substituting in succession the value of m an- 
swering to the different colors for water, we shall 
have values for 9>,and consequently for </, equation 
(93), which substituted in equation (92), will give 
the angles subtended by the radii of the colored 
arches which make up what is called the 'primary 
bow. 
For red, m — 1,3333, hence 
cos <p =. ’5092 =* cos 59° 21' 
sin cp = v^i_cos2 <p — 8603; 
this last in equation (93), gives 
ip 40’ ir 170 
THE RAINBOW. 
and these values of ? and <p' in equation (92), give 
.5 = —118°. 42' + 160°. 44 = 42°. 02. 
For the violet, m = 1,3456, 
cos cp =» -5199 =* cos 58°. 4l£ 
sin <p = -8543 
cp' =. 39°. 25‘ 
3' ~ _ 117°. 23' + 157°. 40 = 40°. 17, 
hence, the width of the primary bow is 
3 — V = 42°. 02'— 40°. 17 = 1®. 45. 
If there be two internal reflections, as in fig. (78), 
we shall, by making n — 2, find 
Vm2— 1 
— 
8 
and obtain, by a process entirely similar, the ele- 
ments of what is called the secondary bow. THE RAINBOW. 
171 
For the red rays, 
<5 — 50°. .57' 
violet, 
8' - 54°. 07' 
and 
s'—s- 3a 10 • 
the value of 8' in the secondary, being greater 
than 8, the violet will occupy the outside, and 
the colors, therefore, be arranged in an order 
the reverse of that in the primary. Taking 
the difference between the values of 8 in the pri- 
mary and secondary bows, we will obtain the 
space between them, which is 50°. 57'—42°. 02' 
= 8°. 55'. The solar disk being about 32', the 
width of both bows must be increased by this 
quantity, the solution having been made upon the 
supposition that the light flows from a point. The 
primary is, therefore, 2°. 17' in width, and the 
secondary 3°. 42'. The half of 32' being added 
to the radius of the red in the primary, will give 
42°. 18', hence, if the sun be more than that 
height above the horizon, this bow cannot be seen. 
When more than 54°. 23', no part of the seconda- 
ry will be visible. 
By substituting in equation (95), 3 for n, we 172 
THE RAINBOW. 
might find the radii of a third bow, which would 
be found to encircle the sun at the distance of 
about 43°. 50'; but the proximity of the sun, 
together with the great loss of light arising from 
so many reflections, renders this bow so faint as to 
produce no impression ; it is, therefore, never 
seen. 
Differentiating equation (94), and reducing, it 
becomes 
d* 8 , 2(rc-J-l) sin (cp1—q>) x 
dcp2 V m cos2 cp' ’ / 
and since <*>'o, the sign of sin (<*>'—?) will be ne- 
gative, and hence, 8 was a maximum for the pri- 
mary and a minimum for the secondary. This 
explains the remarkable fact, (fig. 79), that the 
space between these bows always appears darker 
than any other part of the heavens in the 
vicinity of the bow, for no light twice refracted 
and once reflected can reach the eye till the 
drops arrive at the primary ; and none which is 
twice refracted and twice reflected, can arrive at 
the eye after the drops pass the secondary ; hence, 
while the drops are descending in the space be- 
tween the bows, the light twice refracted with one 
and two intermediate reflections, will pass, the 
first above, and the second below or in front of 
the observer. THE RAINBOW. 
173 
The same discussion will, of course, apply to 
the lunar rainbow which is sometimes seen. 
100. Luminous and colored rings, called halos, 
are occasionally seen about the sun and moon ; 
the most remarkable of these are generally at dis- 
tances of about twenty-two and forty-five degrees 
from these luminaries, and may be accounted for 
upon the principle of unequal refrangibility of 
light. They most commonly occur in cold cli- 
mates. It is known that ice crystalizes in minute 
prisms, having angles of 60°, and sometimes 90°; 
these floating in the atmosphere constitute a kind 
of mist, and having their axes in all possible di- 
rections, a number will always be found perpendi- 
cular to each plane passing through the sun or 
moon, and the eye of the observer. One of these 
planes is indicated in (fig. 80). 
S m being a beam of light parallel to S E, drawn 
through the sun and the eye, and incident upon the 
face of a prism whose refracting angle is 90° or 
60°, we shall have the value of d, corresponding 
to a minimum from equation (8), by substituting 
the proper values of m for ice. The mean value 
being 1.31, we have 
sin (<? -f 60) = 1.31 . sin 30° 
* <J = 40°. 55'. 10" — 30° = 10° 55.10 
5 =21°. 50'. 20” 174 
INTERFERENCE OF EIGHT. 
and 
sin £((3 -j-:90) = 3.31. sin 45°. 
y $ =:67°. 52' — 45 = 22.. -52 
S — 45°. 44'; 
Other phenomena of a similar nature will be 
noticed hereafter. 
Interference of Light. 
ML Let AB, A' B' (fig:. 81), ibe two parallel 
linear radiants, such as may be formed by cylin- 
drical lenses very near each other, from which 
proceed diverging rays of light. A portion of the 
rays from «we of these radiants will cross a por- 
tion cof those from the other, and if the whole be 
received npon a screen M N, parallel to the plane 
<of the radiant, there will be an illuminated space 
comsistingef a central portion due to the rays from 
both radiants, and two external portions, each en- 
lightened by the rays from only one. So far as any 
property «af light thus far noticed is concerned, we 
should expect to find the central portion of uni- 
form brightness and <of double the intensity of the 
•others. Such, however, is not the case ; for when 
the radiants are very near each other, or the an- 
gle smbtended by the d istance between them at the INTERFERENCE OF LIGHT. 
175 
distance of the screen is very small, this space is 
found to be covered by alternate stripes of light and 
total darkness, the stripes being parallel to the radi- 
ants. These stripes are confined to the central 
space, and a bright one passes directly through the 
centre, the others alternating at distances on one 
side exactly equal to those which mark the alterna- 
tions on the other; so that for each stripe on the right, 
for example, there will always be found a similar 
one at the same distance on the left. This will 
always be the case, no matter how the experiment 
be performed, provided the foregoing conditions 
be fulfilled ; and hence it is obvious, that the phe- 
nomenon must depend upon some property or affec- 
tion of light itself. If the light from one of the 
radiants be arrested by the interposition of some 
opaque substance, the stripes will disappear, and 
the illuminated space in the screen will become of 
one uniform brightness equal in intensity to that 
due to the other radiant; from which it follows, 
that the rays from both radiants are necessary to 
produce the stripes. 
Considering, now, the paths of the crossing 
rays, it is manifest that the central bright stripe is 
equally distant from both radiants, and that the 
distances of any other stripe, whether bright or 
dark, from the radiants will be unequal, the differ- 
ence depending upon the distance of the stripe 
from the centre. Hence it is inferred, “ that any 
ray divided along its length into intervals equal to 
these differences, must at the alternate points be 176 
INTERFERENCE OF LIGHT. 
somehow in a different state; such that if two rays 
cross at points where they are in the same condi- 
tion, they conspire to form a bright point; and if 
in different conditions, they neutralize and destroy 
each other, and leave a point of darkness.” 
This is called the interference of light. 
If instead of receiving the light upon a screen, 
it be permitted to fall on a convex lens provided 
with a micrometer, by looking through the lens, 
the stripes will not only appear to greater advan- 
tage, but the distances between them may be ac- 
curately measured as well as the angular distance 
of the radiants ; and hence the intervals along the 
rays at which the different states occur, may be 
determined. 
To do this, it is in the first place obvious, that 
the points of the rays diverging from either ra- 
diant which are in the same condition, will be on 
the surfaces of concentric cylinders whose com- 
mon axis is the radiant line ; the surfaces about 
one radiant will intersect those about the other, 
and if the diverging rays be taken within very 
narrow limits, these surfaces may, within such 
limits, be regarded as planes whose intersec- 
tions with a plane normal to the radiants will be 
right lines perpendicular to the rays. Thus A A' 
(fig. 82), being the points of intersection of the 
radiants with the normal plane, and A B, A' B, any 
two rays ; then M N, M' N', mn, m! ri, which re- 
present the intersections of the plane normal to 
the radiants A A' with the cylindrical surfaces, INTERFERENCE OF LIGHT, 
177 
may be regarded as right lines perpendicular to 
their respective rays : hence, calling l the dis- 
tance between two consecutive cylinders, c the 
the distance O O' on the screen or in the micro- 
meter, between the central and next bright 
stripe in order, and 2 v* the angle subtended 
by the line joining the radiants, we get, consider- 
ing B O equal to ~ i, 
A = 2 c . tan y (96) 
hence, 
t = £ l. cot y. 
and for the distance of any bright stripe from the 
centre 
nc = c' = .... (97) 
from which it follows, i being constant, that the 
stripes will be diminished as is increased ; a re- 
sult confirmed by experience, for if the screen be 
made to approach the radiants, in which case v 
will increase, the stripes become narrower and ex- 
tend over a much smaller space ; again, if the 
radiants be made to separate, the same effect is 
observed till the stripes become so fine as not to 
be perceptible, INTERFERENCE OF LIGHT. 
102. The law according to which any one 
stripe, the nth for example, approaches the central 
bright stripe, when the screen is moved towards 
the radiants, is readily ascertained. This nth 
stripe is formed by the union of two rays that dif- 
fer in length by some multiple n i of and both 
rays being diminished successively by 1,21, 3 A, &c., 
the circles described with these diminished radii will 
intersect and determine the points between the 
screen and radiants at which this stripe will occur; 
and the locus of these intersections will be an 
hyperbola, since the difference in the lengths 
of the rays corresponding to its point is con- 
stant. Hence, the co-ordinates SP (fig. 83), of 
this curve referred to the line B B', connecting the 
central bright stripe with the middle point be- 
tween the radiants, regarded as an origin, will de- 
termine the law in question. 
103. If the linear radiants become radiant 
points, the stripes upon the screen will assume the 
form of diverging hyperbolas. 
104. When the radiant points (fig. 84), are 
taken at unequal distances from, and in the same 
line perpendicular to the screen, the stripes will 
take the shape of concentric circles around the 
point in the screen at which it is pierced by the 
line joining the radiants. INTERFERENCE OF LIGHT. 
179 
105. It follows from the existence of these 
similar and dissimilar points at equal intervals 
along any ray of light, that if two rays occupy the 
same position throughout their entire lengths, and 
the points of the one coincide with the similar 
points of the other, they will produce double the 
effect due to one of them separately ; but if the 
dissimilar points coincide, they will destroy each 
other, and be rendered incapable of producing 
any effect whatever. If intermediate points should 
coincide, the effect will be intermediate between 
those just stated. 
106. When the experiment of number (101) is 
made with the different kinds of homogeneous 
light, the stripes for the same position of the radi- 
ants and screen are found to vary, being broad- 
est in red and narrowest in violet; hence the in- 
tervals along the ray at which the different states 
occur will not, equation (96), be the same for the 
different colors. 
The following table, which is the result of very 
accurate experiments, exhibits the value of x for 
the various colors of the spectrum. INTERFERENCE OF LIGHT. 
Colors. 
Length of 
X 
in parts of an inch. 
Number of 
X 
in one inch. 
Extreme red, 
0.0000266 
37640 
Orange, 
0.0000240 
41610 
Yellow. 
0.0000227 
44000 
Green, 
0.0000211 
47460 
Blue, 
0.0000196 
51110 
Indigo, . 
0.0000185 
54070 
Extreme violet, 
0.0000167 
59750 
and from which we may infer, 
First. That when the experiment is performed in 
common light, the central white stripe results from 
the superposition of the various stripes of red, 
orange, and so on to violet; and because these 
elementary stripes are not of the same width, the 
central stripe will have an extremely small portion 
of color on its edges, red being on the outside on 
account of the greater width of this color. 
Second. Leaving the central stripe and proceed- 
ing to the right and left, the amount of colors on 
the borders of the stripes will continually increase 
till they exhibit several colors in the order of the 
spectrum, the red being always at the greatest 
distance from the centre. 
Third. The dark spaces a little beyond this, 
will be encroached upon by the further separation 
of the elementary colors till they disappear, and 
the whole system of stripes be lost. 
107. Thus far the rays have been supposed to INTERFERENCE OF LIGHT. 
181 
proceed from the radiant to the screen through the 
atmosphere ; but if the experiment be performed 
in a medium of greater refractive power, the 
stripes will be reduced in width and crowded into 
a narrower space, and consequently the intervals 
along the rays at which the different states occur, 
will be shortened ; and this effect is found, ac- 
cording to very accurate experiment, to be exactly 
proportional to the index of refraction of the 
medium as referred to atmospheric air. 
Returning to the experiment in air, if a very 
thin plate of glass be interposed in front of one 
of the radiants, and parallel to the plane of the 
radiants, the whole system of stripes will be 
shifted towards the side of the interposed glass. 
If an exactly similar plate be placed in front of the 
other radiant, and parallel to the first plate, the 
stripes will be restored to their original position. 
If one of the plates be slightly inclined, so as to 
cause the pencil passing through it to traverse a 
greater thickness, the stripes will all move towards 
that side, and by gradually increasing the inclina- 
tion, the stripes will pass entirely out of the bright 
space illuminated by both radiants, and thus dis- 
appear. 
Taking plates of any other medium, possessing 
a greater refractive power than glass, and of the 
same thickness as before, it is found that the 
effects just noticed will be increased, and in the 
direct ratio of the refractive indices. 
In the shifting of the stripes, it is evident that 
the Jengths of the rays producing the central one, 182 
INTERFERENCE OF LIGHT. 
are rendered unequal, and that the differences as 
to length existing among the rays that form the 
other stripes, are not the same as before. It is 
now proposed to investigate this change. 
For this purpose, let the rays from both radiants 
pass through a prism of any medium, as glass, 
having a very small refracting angle = i, (fig. 85), 
the first face being held parallel to the plane of the 
radiants. The thickness of the prism traversed 
by two interfering rays will be different; call this 
difference, which is rn in the figure, d. Since 
the angle 2 V7, made by the same rays, is very 
small, they will enter the first surface under very 
small angles of incidence, and both being refrac- 
ted towards the perpendicular, their direction 
through the prism will be nearly normal to that 
surface; hence, denoting by b the distance rn' 
between the rays measured on the second surface, 
we have 
d = b . sin 
but under the above supposition, the angle of inci- 
dence at the second surface will be equal to i; and 
denoting the corresponding angle of refraction 
by (p, we also get 
sin (p = m sin i; 
with S, the new position of the central stripe, as a INTERFERENCE OF LIGHT. 
183 
centre, and radius Sr, describe the arc rr ; r‘ n! 
will be the difference of the interfering rays after 
leaving the prism. Calling this d', and because 
rr' is very small, we have 
d' <*= b . sin cp — b m . sin i =* vi d 
hence, 
d : d' :: 1 : m 
that is, the difference of lengths within the prism, 
is to that out of the prism, as unity is to the index 
of refraction. But the number of intervals 
i in d while in the prism, is to the number 
in an equal length in the air, as the index of re- 
fraction is to unity, and hence, the number of in- 
tervals at which the different states occur will be 
the same in both rays, reckoning from the radiants 
to the point of interference; and for any other 
stripe, although the interfering rays will present a 
greater or less linear difference than before the 
prism was interposed, yet the difference in the num- 
ber of intervals will be accurately the same. 
Whence it follows, that if two rays interfere and 
form any given stripe with a given difference of 
route, two other rays, having a greater or less dif- 
ference will form the same stripe, provided one of 
the rays pass through a medium of such refractive 184 
INTERFERENCE OF LIGHT. 
power as to make their difference, as estimated by 
the number of intervals, the same as before. 
This fact of a greater or less number of inter- 
vals occurring within the same length of a ray in 
its passage through media of greater or less re- 
fractive power, is called, in the first case, the re- 
tardation, and in the second, the acceleration of 
the intervals. 
The whole difference in the lengths of the in- 
terfering rays will be 
d — — 1) 
and returning to the parallel plate whose thickness 
will be denoted by t, we shall have by putting n i} 
for d!— d, 
nl = t(m — 1). 
Substituting this value in equation (97), it gives 
. . (98) 
Taking homogeneous light* and considering only DIVERGENCE OF LIGHT. 
185 
the extreme rays of the spectrum, viz: red and 
violet, denoted by r and v, we have 
' 4! n cot^ 
Cr *== i 1). g . 
' 4, t\ COtV' 
C p = t (171d 1). 1 
and by subtraction 
, cot V7 
c r — c v = t (mr — mv).—-— . (99). 
i 
When the first member exceeds the width of a 
stripe, the colors encroach upon the dark spaces, 
and the stripes will disappear. This is in perfect 
accordance with experience, for if the thin plate of 
glass be replaced by a thick one, the stripes will 
vanish. 
Divergence of Light. 
108. It has been stated (12), as a general fact, 
that the motion of light through a medium of 
homogeneous density, is rectilinear ; to this, how- 
ever, there is a remarkable exception. It is found COLORED FRINGES OF 
that when light passes near the edge or boun- 
dary of any body, though it continue in the same 
medium, it has a tendency to, and in a certain 
degree actually does, diverge anew from this boun- 
dary as a new origin. This is rendered evident 
by examining either the shadows of opaque bodies 
in diverging pencils of light, or the portions of 
these pencils transmitted through small openings 
and received upon a screen ; in the first case the 
shadows will be less, and in the second the illu- 
minated space will be greater than would result 
were the rays to pass in straight lines tangent 
to the bodies or edges of the openings. The 
pencil being accurately formed by a convex lens, 
and the dimensions of a small body placed in it 
being known, the fact of new divergence may be 
made matter of observation and distinct measure- 
ment. 
Colored Fringes of Chadows and Apertures. 
109. When an object is placed in a pencil, such 
as may be formed by admitting light through a 
very small aperture into a dark chamber, or by a 
convex lens, and the shadow of the object is re- 
ceived upon a screen, it is found to be bordered 
externally by bands, usually three in number at 
decreasing distances from each other, each band 
being made up of different colors. These bands 
are parallel to the outlines of the shadow, except SHADOWS AND APERTURES. 
187 
when the latter terminate in a salient angle, in 
which case, the bands curve around it; or when 
the outlines form a re-entering angle, when the 
bands will cross and run up to the shadow on each 
side. 
In white light, the bands, reckoning from the 
shadow, are black, violet, deep blue, light blue, 
green, yellow and red ; blue, yellow and red ; pale 
blue, pale yellow and pale red. 
In homogeneous light, the bands increase in 
number and are alternately dark and bright. In 
passing from one color to another, they vary in 
width, being broadest in red and narrowest in vio- 
let ; and it is from the partial superposition of 
these and the remaining colors, that the different 
colors arise when the experiment is made with 
white light. 
These bands are entirely independent of the 
nature and figure of the body whose shadow they 
surround, being the same when formed by a mass 
of platina or a bubble of air—by the back or edge 
of a razor. 
The shadow being received upon a convex lens, 
behind which is placed a micrometer, the linear ele- 
ments of the fringes may be measured to any desired 
degree of accuracy. These measurements show 
1st. That the distances between the fringes and 
the shadow diminish as the lens approaches the 
body, and finally vanish, so that the fringes have 
their origin close to the edge of the body. 188 
COLORED FRINGES OF 
2d. That the locus of each fringe is an hyper- 
boloid of revolution, terminating near the edge of 
the body. 
3d. That, the distance of the lens from the 
body remaining the same, the fringes will be more 
dilated, as the body approaches the luminous point. 
It is also found, that when the luminous 
point is increased so as to become an appreciable 
circle, the bands formed by the light proceeding 
from each of its points overlap and confuse each 
other, obliterating the colors, and forming a 
penumbra, wdiich consists of a ring whose bright- 
ness varies from the edge of the shadow, where 
it is least, to its exterior boundary where it is 
greatest. 
110. If the size of the body be much reduced 
in one direction, parallel to the screen or plane of 
the lens, the shadow will be found to consist of 
bright and dark stripes, parallel to the length of 
the body, a bright stripe occupying the centre. If 
the body be small and of a circular form, having 
its plane parallel to that of the screen, the 
shadow will be made up of a series of concen- 
tric bright and dark circles, having a bright spot 
in their centre. 
As the body diminishes in size, the stripes dimin- 
ish in number and increase in width, till all dis- 
appear but the central illumination. The re- 
verse effect will arise either on increasing the size SHADOWS AND APERTURES. 
189 
of the body, or diminishing its distance from the 
screen. 
111. If a portion of the pencil be transmitted 
through a small and well defined circular aperture 
and received upon a screen, concentric rings 
will also be produced; and if the transmitted 
portion be viewed through a convex lens, the 
hole will appear as a bright spot, encircled by 
rings of the most vivid colors, which undergo a 
great variety of changes, both as regards tint and 
linear dimensions, in varying the distance of the 
lens from the aperture, and that of the aperture 
from the radiant or luminous point. 
When the light is transmitted through two very 
small apertures, very close together, rings corres- 
ponding to each will be formed as before, and in 
addition there will be found a number of straight 
parallel fringes between the centres of the circles, 
and at right angles to the line joining them ; two 
other sets of parallel fringes will also be seen in 
the form of St. Andrew’s cross proceeding from 
the space between the centres ; and by multiplying 
the number of the apertures and varying their re- 
lative dimensions, a set of phenomena arise of ex- 
ceeding brilliancy and beauty. 
112. The colored fringes of shadows and small 
apertures, as well as all appearances referred to 
under this head, are explained upon the principle 190 
DOUBLE REFRACTION. 
of interference ; the interference taking place be- 
tween the rays of the original pencil that diverge 
anew from the edges of the body or aperture, and 
between these rays and those that experience no 
change in their course. 
These phenomena were at one time called the 
inflection or diffraction of light, and were sup- 
posed to arise from some peculiar action exerted 
by the edges of bodies on the rays as they passed 
near them. 
If the refractive power of the medium in which 
the experiments are performed be increased, the 
phenomena indicate a diminution in the intervals 
along the rays in the same ratio. 
Double Refraction. 
113. In treating of the transmitted portion of 
a beam of light incident upon any deviating sur- 
face, we have heretofore supposed all its rays to 
pursue a single course through the medium indi- 
cated by the constant ratio of the sine of inci- 
dence to that of refraction. This is not, however, 
always the case, there being a large class of bodies 
that possess the power of dividing the intromitted 
beam into two portions, each of which pursues, 
without further subdivision, a rectilinear path de- 
termined by its own peculiar law. This class em- 
braces all crystallized media except those whose DOUBLE REFRACTION. 
191 
primitive form is the cube, the octohedron, and the 
rhomboidal dodecahedron; all animal substances 
among whose particles there is a tendency to 
regular arrangement; and, in general, all solids in 
a state of unequal compression or dilatation. 
The phenomenon presented by any medium of 
dividing a beam which enters it into two parts is 
called double refraction. One of the most re- 
markable bodies in this respect is Iceland spar, 
which is a carbonate of lime, and is found in crys- 
tals of various shapes, all of which may be reduced 
by cleavage to a regular obtuse rhomboid, this be- 
ing its primitive form, or the form of its ultimate 
molecules. This rhomboid has six acute and two 
obtuse angles, and the diagonal A B (fig. 86), 
joining the latter, which is the shortest that can 
be drawn in the figure, is called the axis of double 
refraction or optical axis of the crystal; it is 
so called, because when the beam, on entering the 
crystal, moves along this line, no double refrac- 
tion is observed. But this rhomb may be divided 
into an indefinite number of similar elementary 
rhombs each having an axis parallel to A B, it 
would, therefore, follow that whenever the refrac- 
ted beam takes a direction parallel to A B, there 
should be no double refraction; and this is con- 
firmed by observation. By an axis of double re- 
fraction, we are to understand, therefore, merely a 
direction, determined in the present case by join- 
ing the obtuse solid angles of the rhomb. 
Whenever the beam takes a direction different 192 
DOUBLE REFRACTION. 
from the axis, it is found to be divided into two 
portions, one being refracted according to the or- 
dinary law of the sines, the other according to 
some extraordinary law yet to be explained ; the 
first portion is called the ordinary ray, the other, 
the extraordinary ray. 
114. Any plane drawn through the crystal, 
parallel to the axis, is called a plane of principal 
section, and the peculiar property of such a plane 
is, that whenever it contains the ordinary ray, it 
will also contain the extraordinary ray ; in other 
words, if a principal section, drawn through the 
point of incidence, coincide with the plane of in- 
cidence, it will contain both rays. Whenever 
these planes are inclined to each other, the extra- 
ordinary ray will not be in the same plane with 
the incident and ordinary rays, except in the case 
where the incident plane is perpendicular to the 
axis and consequently at right angles to every 
principal section possessed by the crystal, when 
the incident, ordinary and extraordinary rays will 
again be found in the same plane. 
Let M N (fig. 87), represent a principal section 
of Iceland spar, normal to the upper face of the 
crystal, A B the optical axis, SP a beam of 
light, in the plane of principal section, incident at 
P. Now, if from the point P, the axis P X be 
drawn, the following are the facts determined by 
observation. 
When the incident beam S P is normal to the DOUBLE REFRACTION. 
193 
upper surface, the ordinary ray P o will pass 
through without deviation, but the extraordinary 
ray will be thrown from the axis in the direction 
P e, making the angle oP e upwards of six de- 
grees. When the incident beam takes the position 
S" P, the ordinary ray P o" will be refracted ac- 
cording to the ordinary law with a constant index 
of 1.654, and the extraordinary ray will take the 
direction P e!\ making the angle o" P e" greater 
than oP e, and this angle will reach its maximum 
limit when the extraordinary ray is at right angles 
to the axis. If the incident beam assume the 
position S' P, the ordinary ray will retain its con- 
stant index taking the direction P o', and the ex- 
traordinary ray the direction P e', making the 
angle e' P o', less than e P o. Finally, when the 
incident beam S'" P has a position such that the 
ordinary ray will be refracted along the axis, there 
will be no double refraction, or the angle e' P o' 
will reduce to zero. 
115. Had rock crystal, which occurs in the form 
of hexagonal prisms terminated with six sided 
pyramids been selected, what has been said of 
Iceland spar, would equally apply to it, except that 
the extraordinary ray is always found between the 
ordinary ray and the axis passing through the 
point of incidence, as if drawn towards the lat- 
ter ; and this circumstance has given rise to a di- 
vision of doubly refracting substances into two 194 
DOUBLE REFRACTION. 
classes distinguished by their axes, which are said 
to be positive when the extraordinary ray is be- 
tween the ordinary ray and the axis, as in the case 
of rock crystal; and negative, when the positions 
of these rays are reversed with respect to the axis, 
as in Iceland spar. 
116. If we now suppose, for the purpose of ex- 
planation, the angle of incidence, S P P7 (fig. 88), 
to remain the same, the ordinary ray P O will be 
stationary, no matter what the position of the opti- 
cal axis, since they are independent of each other. 
Not so, however, with the extraordinary ray P e; 
this will, from w'hat has been said, assume, in the 
negative crystal, some direction as P e7, when the 
axis takes the position PX'; and since the index 
of refraction for the extraordinary ray is the ratio 
of the sines of the angles P7 P S and P77 P e', this 
index is variable and less than that for the ordi- 
nary ray in every position of the axis except P O, 
when they will be equal; it will, of course, have 
its minimum value when the extraordinary ray and 
axis are at right angles. The reverse will be the 
case in crystals with a positive axis. 
117- Having ascertained by experiment the 
value of the ordinary index, which will be repre- 
sented by m0, and the maximum or minimum 
value of the extraordinary index, according as the 
crystal has a positive or negative axis, which will DOUBLE REFRACTION. 
195 
be represented by (me), it has been determined 
from theoretical considerations, and confirmed by 
experiment, that all values of the extraordinary in- 
dex between m0 and ?ne may be found by the 
following law. 
O 
Let an ellipsoid of revolution (fig. 89), be con- 
ceived, having its centre C at the point of inci- 
dence, its axis of revolution coincident with the 
optical axis of the crystal, and its polar to its equa- 
torial radius in the inverse ratio of the minimum 
and maximum values of the extraordinary index of 
refraction ; then, in all positions of the extraordi- 
nary ray, its index is equal to the reciprocal of its 
length contained between the centre and surface 
of the ellipsoid. 
The equation of a section of this surface 
through the axis, referred to the centre, is 
A2 y2 -f- B2 X1 — A2 B2, 
and calling r, the length of the extraordinary ray 
between the centre and the surface, and e, its an- 
gle of inclination with the optical axis, it reduces to 
A B _ A B 
V\2sin2 0-{-b2cos‘0 ''^b'2-P(a2—B2)sin'0> 
denoting by m*,, the value of the extraordinary 196 
DOUBLE REFRACTION. 
index sought, we have 
“•=T= V/y+(i-ir>in’fl . . . (1°0) 
in which 
1 
m0 
I 
B = . 
mt 
It is obvious that the coefficient of sin2 o is 
positive or negative according as the axis is posi- 
tive or negative ; hence, the coefficient of sin2 0 de- 
termines the nature of the crystal. 
118. To determine the value of m0 and me, in 
any particular instance, it is in the first place 
known that the index of the extraordinary ray will 
be constant and equal to its maximum or minimum 
value, according to the nature of the body, when re- 
fracted in a plane at right angles to the optical axis ; 
it is only necessary, therefore, to convert the crys- 
tal, by grinding, into a prism whose refracting 
faces shall be parallel to the axis, when both the 
ordinary and extraordinary index may be ascer- 
tained by the method explained in (25). To dis- 
tinguish between the rays, it will, in general, be 
sufficient to move the prism so as to give the POLARIZATION OF LIGHT. 
197 
plane of incidence a slight inclination to its length, 
as in that case the extraordinary ray will be thrown 
out of this plane, and thus become known. 
In Iceland spar 
m0 — 1.6543, 
me — 1.4833; 
hence, 
A = 0.60449, 
B = 0.67417; 
the ellipsoid is, therefore, oblate; and the coef- 
ficient of sin2 d, negative. Tourmaline, beryl, 
emerald, apatite, &c., also belong to this class. 
Quartz, ice, zercon, oxide of tin, &c., give the co- 
efficient of sin2 Q positive ; they are, therefore, of 
the positive class, and the ellipsoid is prolate. 
119. Among doubly refracting crystals there are 
very many that possess two axes of double refrac- 
tion, but in all such cases it has been ascertained 
that there is, in fact, no ordinary ray. 
Polarization of Light. 
120. When a beam of light is incident upon 
any deviating surface, it has been before remarked 198 
POLARIZATION BY REFLECTION. 
that a portion is always reflected and another 
transmitted ; and the relative intensity of these 
will be constant so long as the surface and angle 
of incidence remain the same, no matter to which 
side of the beam the deviating surface be presen- 
ted, provided, the light be in the state in which it 
comes from the sun or any self luminous body. 
But with light that has already undergone some 
reflection, refraction, or other action of material 
bodies, this uniformity of result will not obtain. 
Such light is found to have acquired different pro- 
perties on different sides, for the intensity of the 
reflected and transmitted portions are found mate- 
rially to depend on the side of the beam to which 
the deviating surface is offered. A beam or ray, 
distinguished by this, and other circumstances to 
be noticed hereafter, is said to be polarized. 
Polarization by Reflection. 
121. The intensity of the reflected portion of a 
beam of light, is found to be greater in proportion 
as the refractive index of the medium, and angle 
of incidence are greater. It is, moreover, ascer- 
tained that when reflection from any transparent 
medium takes place under a certain angle of inci- 
dence, called the polarizing angle, the reflected 
beam loses almost entirely the power of being 
again reflected when the reflector is presented in 
a particular manner. POLARIZATION BY REFLECTION. 
199 
M N, and M7 N7 (fig. 90), representing two 
plates of glass, mounted upon swing frames, at- 
tached to two tubes A and B, which move freely 
one within the other about a common axis, let the 
beam S D, from any self luminous body, be re- 
ceived upon the first under an angle of incidence 
equal to 56° ; reflection will take place according 
to the ordinary law in a plane normal to the re- 
flecting surface ; and if the reflected beam D D7, 
which is supposed to coincide with the common 
axis of the tubes, be incident upon the second re- 
flector under the same angle of incidence, the re- 
flector being perpendicular to the plane of first re- 
flection, it will be again reflected in the same man- 
ner as before. 
But if the tube B be turned about its axis, the 
tube A being at rest, the angle of incidence on 
the glass M7 N7 will remain unchanged, yet the 
portion reflected from it will become less and less, 
till the tube B has been turned through an angle 
equal to 90°, as indicated by the graduated circle 
C, on the tube A, when the beam will almost totally 
disappear, or cease to be reflected. Continuing 
to turn the tube B, the reflection from M7 N7 will 
increase till the angle is equal to 180°, when the 
plane of first reflection will be again perpendicu- 
lar to M7 N7, and the whole beam will be reflected; 
beyond this, reflection will diminish till the angle 
becomes 270°, when the beam will be again lost; 
after passing this point, the lost beam will be 200 
POLARIZATION BY REFLECTION. 
gradually restored, till the tube is revolved through 
360°, when the restoration will be complete. 
It thus appears that a beam of light reflected 
from a plate of glass under an angle of incidence 
equal to 56°, immediately acquires opposite pro- 
perties, with respect to reflection, on sides dis- 
tant from each other equal to 90°, measuring 
around the beam ; and the same property at dis- 
tances of 180°. 
We have supposed the angle of incidence 56°, 
if it were less or greater than this, similar effects 
would be observed, though less in degree ; or, in 
other words, the beam would appear but partially 
polarized, the palarizing effect decreasing as the 
angle of incidence recedes from that of polariza- 
tion, being nothing at the incidence of zero and 
90°. 
The plate M' N' is called the analyzer; the 
plane of first reflection is called the plane of polar- 
ization, and the beam is said to be polarized in 
this plane. The position of this plane in any 
polarized beam may readily be ascertained by the 
total reflection which takes place from the analy- 
zer when the latter is perpendicular to it. Start- 
ing from this position of the analyzer with respect 
to the plane of polarization, and calling d, the 
angle between the plane of polarization and that 
of second incidence, which is equal to the angle 
through which the analyzer has at any time been 
turned about the first reflected or polarized beam; 
A, the intensity of this beam, and I, the variable POLARIZATION BY REFLECTION. 
201 
intensity of that reflected from the analyzer in its 
various positions, it has been conceived, on careful 
investigation, that in uncrystalized media the 
formula 
i = a cos2 d , 4 , 4 . (101) 
will express the law according to which a 
ized beam will be reflected from the analyzer when 
the angle of incidence is equal to that of polari- 
zation. 
According to this law, if we conceive a common 
beam, as it emanates from any self-luminous body, 
to be composed of two beams polarized in planes 
at right angles to each other, we should have, call- 
ing I and F the intensity of the reflection in the 
first and second respectively, 
i -f i' = acos2 a-f- a . cos2 (90° — a) — A 
or the intensity of the reflected beam will be the 
same on whatever side of the incident beam the 
analyzer is presented. 
118. What has been said of the effects of glass 
on light is equally true of othef transparent media, 
except that the polarizing angle, which is constant 
for the same substance, differs for different bodies, 202 
POLARIZATION BY REFLECTION. 
Sir David Brewster discovered, from very nu- 
merous observations, that the tangent of the maxi- 
mum polarizing angle is always equal to the re- 
fractive index of the reflecting medium taken in 
reference to that in which the ray is reflected: 
thus, calling the relative index m, and the polari- 
zing angle <p, we shall have, 
tan <p = to. . . . . . . (102) 
Example. Let it be required to find the polari- 
zing angle when light is moving in water and re- 
flected from glass. The refractive index for water 
and glass are 1,336 and 1,525, respectively, 
hence, 
1.525 , 
m==T336_== 11415 =tan<p 
or 
9 = 48°. 47'. 
If the refractive indices of the media were* 
equal, we should have 
m = 1 
and 
9 =“* 45°. POLARIZATION BY REFLECTION. 
203 
The following are the values of ?, for the dif- 
ferent substances named, the ray being reflected 
in air. 
Water, 
53°. 11' 
Crown glass, 
56°. 55' 
Plate glass, 
57°. 45 
Oil of Cassia, 
58°. 39 
Diamond, 
68°. 6' 
123. It is obvious that according to the law ex- 
pressed by equation (102), there can be no such 
thing as perfect polarization by reflection in a 
beam of white light, since the refractive index is 
not the same for the different colors; and hence 
there can never be total absence of light at the 
analyzer; but a certain tint will be reflected, 
whose intensity will depend upon the dispersive 
power of the medium. For bodies of very high 
refractive powers, which are also, in general, 
highly dispersive, we must, therefore, understand 
by the polarizing angle, that angle of incidence at 
which the reflected beam approaches nearest to 
perfect polarization. This angle being ascer- 
tained for opaque bodies by experiment, the rela- 
tion expressed by equation (102), furnishes the 
means of ascertaining their refractive indices. 
Thus, the maximum polarizing angle for steel is a 
little over 71°, the natural tangent of which is 
2.85, which is, therefore, according to the law, its 
refractive index: the polarizing angle for mercury 204 
POLARIZATION BY REFLECTION. 
is about 76°. 30', and its index, consequently, 
4.16. 
With metallic substances, no complete polariza- 
tion takes place, 
124. We have spoken, thus far, only of the 
action at the first surface of the glass plate ; it is 
found that the light reflected at the second surface 
is as perfectly polarized as that reflected at the 
first, and in the same plane, when the faces of the 
plate are parallel. This is a consequence of the 
same law for, (fig. 91), 
sin cp sin cp 
m = tan cp = = —7 
cos cp sin cp 
hence, 
cos cp =» sin cp' 
or <p' is the complement of 9>, and the first reflec- 
ted beam is perpendicular to the first refracted. 
Moreover, 
1 1 
-— = — cot cp = tan cp 
m tan <p 
but — is the index of the ray passing out of the 
glass; hence v' is the maximum polarizing angle 
for the second surface, POLARIZATION BY REFRACTION. 
205 
If a series of parallel plates be employed in the 
form of a pile, the rays reflected from the second 
surface coming off polarized in the same plane, a 
very intense polarized beam may be obtained. 
This beam will, however, never contain more than 
half the original incident beam, no matter how 
great the number of plates employed. 
125. Although a beam of light is but imper- 
fectly polarized when reflected once at an angle dif- 
fering from that of polarization, yet by repeating the 
reflections a sufficient number of times the polar- 
zaition may be completed; and in doing this it is 
not necessary that the reflections take place at 
the same angle of incidence, but some may be 
above and some below the polarizing angle. In 
general, the number of reflections will increase as 
the angle of incidence recedes from that of polari- 
zation on either side. 
Polarization by Refraction. 
126. When a beam of light S P (fig. 92), is in-r 
cident upon several plates of glass, laid one above 
the other, the transmitted portion P/ S', will be 
found partially polarized in a plane at right angles 
to that of refraction, and the degree of polariza- 
tion will be greater as the number of plates is im 
creased, so that if the number be considerable, the 206 
POLARIZATION BY ABSORPTION. 
ray may be wholly polarized. The polarizing 
effect of the plates will increase with the refrac- 
tive indices of the media of which they are com- 
posed. 
It is ascertained that at all angles of incidence 
upon a plate of glass, the reflected and transmit- 
ted portions of a beam contain equal quantities of 
polarized light, the planes of polarization being at 
right angles to each other. 
Polarization by Absorption. 
127. A plate of Tourmaline, about — of an inch 
thick, cut parallel to the axis, possesses the pro- 
perty of intercepting a part of a beam of light 
and transmitting the other perfectly polarized in a 
plane at right angles to the axis of the crystal. 
If light previously polarized, be incident upon 
the plate with its plane of polarization per- 
pendicular to the axis, it will be wholly transmit- 
ted ; but if parallel, it will be wholly absorbed or 
intercepted. This is another property by which 
polarized light may be distinguished. Hence, two 
plates of tourmaline form a most convenient appa- 
ratus for experimenting with polarized light when 
so arranged as to be capable of turning about a 
common axis, the one being used to polarize 
light, the other to analyze it. Plates of agate and 
some varieties of quartz possess similar properties, POLARIZATION BY DOUBLE REFRACTION. 
207 
Polarization by Double Refraction. 
128. It was stated (114), that a beam of light, 
S P (fig. 98), incident perpendicularly on the 
upper surface of a crystal of Iceland spar, will 
be divided into two parts, one of which P e will 
be refracted according to the extraordinary law, 
and the other P o will pass through without 
deviation. These beams, which will leave the 
crystal in the plane of principal section, nor- 
mal to its upper face, in parallel directions, are 
found to be polarized in planes at right angles to 
each other, that of the ordinary ray being coinci- 
dent with, and that of the extraordinary ray per- 
pendicular to, the plane of principal section. 
If these rays be received upon the upper face 
of a second crystal whose optical axis is parallel 
to that of the first, the beams ee' and oo' it is 
found, will not be again divided, but the whole of 
the latter will undergo ordinary, while the whole of 
the former will experience extraordinary refrac- 
tion ; and if the crystals be of equal thickness, 
the separation or distance e" o", will be double eo. 
If either or both of the beams e e!, o o' had been 
polarized by reflection, refraction or absorption, 
the action of the second prism would be precisely 
the same; this is, therefore, another characteris- 
tic property of polarized light, viz. that it will not 
undergo double refraction when its plane of pola- 208 
POLARIZATION BY DOUBLE REFRACTION. 
rization is either parallel or perpendicular to the 
plane of principal section ; being in the former 
case wholly refracted according to the ordinary, 
and in the latter according to the extraordinary 
law. The reverse would have been the case if 
the crystal, like quartz, had possessed a positive 
axis. 
129. When the crystals M' N' is turned around 
on its base so that the principal sections of the 
crystals, which are normal to the upper surfaces, 
make an angle with each other, each of the beams 
o o' and ed will be again divided into an ordinary 
and extraordinary beam, whose relative intensities 
will depend upon the inclination of the principal 
sections to each other. To avoid complication, 
let us suppose the beam P e to be arrested by 
sticking a piece of wafer to the lower surface of 
the first crystal at e, then will the intensities of the 
portions into which the beam o o' is divided by the* 
second crystal be expressed by the formulas 
Oo = A. cos'2 a ' 
Oe — A . sin2 a ' 
(103) 
Wherein A, represents the intensity of the beam 
o o'; a, the angle made by the principal sections' 
of the crystals ; Oa the intensity of the ordinarily INTERFERENCE OF POLARIZED LIGHT. 
209 
refracted portion ; and Oe that of the portion re- 
fracted according to the extraordinary law. 
Removing the wafer from e, and calling Ee and 
E0 the intensities of the extraordinary and ordi- 
nary beams into which Pe is separated by the 
second crystal, and B its intensity on leaving the 
first crystal, we shall, in like manner, have 
Ee — B. COS2«, * 
E0 — B. sin2 a; | 
. (104) 
taking the sum of the four emergent beams, there 
will result, 
00 -f- Oe -|- Ei -j- Eo — A -f- B. 
The beams O0 and Oe, in equations (103), are 
always found to be polarized, the former in the 
plane of principal section of the second crystal, 
the latter in a plane at right angles to it; and the 
same remark being applicable to E0 and Ee, in 
equations (104), it follows that the planes of polar- 
ization of O0 and E0 will be parallel to each other, 
as also those of Oe and Ee. 
Interference of Polarized Light. 
130. Let abc (fig. 94), represent a thin plate 210 
INTERFERENCE OF POLARIZED LIGHT. 
of Iceland spar, made by sections perpendicular 
to the optical axis. A pencil of common light be- 
ing incident upon it in such direction that its cen- 
tral ray shall coincide with the axis of double re- 
fraction, this ray will pass through without being 
divided ; but not so with the other rays. These, 
taking a direction different from that of the axis, 
will undergo double refraction, and the index of 
refraction for any extraordinary ray, depending 
upon the value of in equation (100), wall vary 
as the ray is taken at a greater or less distance 
from the axis. The acceleration of the intervals 
at w7hich the different states occur on these rays 
in passing through the plate, will, therefore, (107), 
be variable, while that for the ordinary rays will 
be constant. In a positive crystal, the accelera- 
tion for the ordinary rays will be least, and in the 
negative greatest. 
The two classes of rays (fig. 95), being super- 
posed, or nearly so, on leaving the plate, will, in 
consequence of this difference of acceleration, 
be in a condition to interfere ; and since the rays 
are symmetrically disposed about the axis, we 
should expect, on applying the eye to the opposite 
side of the plate, to find that line surrounded by a 
number of concentric circular rings or fringes, whose 
appearance would depend upon the difference in 
the accelerations above referred to, and the thickness 
of the plate within which the cause of that differ- 
ence is permitted to act. No such fringes, how- 
ever, are observed under the circumstances men- INTERFERENCE OF POLARIZED LIGHT. 
211 
tioned, and the reasons are found in the fol- 
lowing laws relating to the interference of polar- 
ized light, which are due to M. M. Fresnel and 
Arago. 
ls£. Two rays polarized in the same plane, in- 
terfere with each other just as natural light. 
2d. Two rays polarized in planes at right 
angles to each other, will not interfere under the 
same circumstances that cause the interference of 
natural rays. 
3d. Two rays primitively polarized in opposite 
planes, (i. e. at right angles to each other), may 
afterwards be reduced to the same plane without 
acquiring the power of interfering with each other. 
4th. Rays polarized in opposite planes, and 
then reduced to the same plane, acquire the power 
of interfering with each other, provided they belong 
to a beam the whole of which had been previ- 
ously polarized in ONE AND THE SAME PLANE. 
5th. In the interference of polarized rays pro- 
duced by double refraction, the place of the colored 
fringes, is not alone produced by the difference of 
route as estimated by the number of inter cals, but 
in certain circumstances a difference of half an 
interval must bj allowed for. 
The first and second of these laws are proved 
by the following experiment. 
M N (fig. 96), represents a thin plate or lamina 
of sulphate of lime, upon which are incident the 
rays of natural light proceeding from two narrow 
slits L, R, very close together, in a thin sheet of 212 
INTERFERENCE OF POLARIZED LIGHT. 
copper. As the sulphate possesses the power of 
double refraction, each pencil, (such we may con- 
sider the collection of rays proceeding from the 
slits), will be divided into two, an ordinary and 
extraordinary one, which will be of equal inten- 
sity, and those marked e, will be polarized oppo- 
sitely to those marked o. Four combinations may, 
therefore, be formed, viz. 
1st. R0) may interfere with L0; 
2d. Re, “ “ Le; 
3d. Ro, “ “ L« ; 
4th. Re, “ “ L0. 
Of these Ru and L0 are similarly polarized, and 
they have been equally accelerated ; therefore, if 
they interfere, they will give rise to a system of 
fringes corresponding to the middle of the space be- 
tween two planes through the slits and perpendicu- 
lar to the sheet of copper. The same may be said 
of Le and Re, and hence these two sets of fringes 
will be superposed and appear as one of double 
intensity. A plane midway between those through 
the slits, will be called the axis plane. 
If R0 and Le, which are oppositely polarized, in- 
terfere, the fringes thence arising must, (107), be 
thrown to that side of the axis plane on which the INTERFERENCE OF POLARIZED LIGHT. 
213 
intervals are most accelerated, the distance from 
this plane depending upon the difference of accele- 
ration and the thickness of the lamina. In like 
manner, on the supposition that Re and L0 inter- 
fere, their fringes would be thrown to the opposite 
side of the axis plane. Only one set of fringes 
is, however, seen, viz : those corresponding to the 
axis plane *, therefore, the rays oppositely 'polarized 
do not interfere. 
But if the plate of sulphate be cut in two, at its 
intersection with the axis plane, and one half be 
turned in its own plane through 90°, Re and L0, 
and Le and R0, will become similarly polarized, 
while the reverse will be the case with R0 and L0, 
Le and Re; and it is, accordingly, found that the 
central fringes will disappear and the lateral 
fringes start into existence ; the half lamina being 
still further revolved, the latter fringes disappear 
and the others return. 
The truth of the 3d law may be thus shown. 
Take two perfectly similar plates of tourmaline, 
and apply one before each slit in the copper, and 
in such manner (123), that the two transmitted 
pencils, denoted by L and R, shall be oppositely 
polarized. These being received upon a doubly 
refracting crystal, whose plane of principal section 
is inclined 45° to their planes of polarization, 
they will each be divided into two of equal inten- 
sity, giving two pairs polarized in the same plane. 
We ought, therefore, expect to find two sets of 214 
INTERFERENCE OF POLARIZED LIGHT. 
fringes, one produced by the interference of the 
ordinary rays from R and L, the other by that of 
the extraordinary rays from R and L, yet no 
fringes are seen. 
The following experiment is mentioned in sup- 
port of the 4th and 5th laws. A lamina of sul- 
phate of lime is presented perpendicularly to a 
polarized pencil, with its plane of principal section 
making an angle of 45° with the plane of polar- 
ization, and immediately behind this is a thin 
plate of copper pierced with two very narrow 
slits, denoted by R and L, near each other, from 
each of which, emerge two equal pencils, 
R o, 
R e, 
Lo, 
L e, 
oppositely polarized, viz : in planes making angles 
of 4-45° and —45° with the plane of primitive 
polarization. A rhomboid of Iceland spar is now 
interposed with its principal section parallel to the 
primitive plane of polarization, and therefore, 
making an angle with that of the lamina of 45°. 
Each of the four pencils is thus divided into INTERFERENCE OF POLARIZED LIGHT. 
215 
two equal pencils, an ordinary and extraordinary, 
giving rise to eight pencils in all, viz. 
R o o, R e o ; 
Loo, Leo; 
Roe, Ree; 
hoe, he e. 
Any corresponding rays of the pencils Ro and 
Re, after quitting the lamina, are parallel, and, when 
the thickness of the lamina is Small, may be regarded 
as superposed; but they have, within the lamina,been 
differently accelerated by a space which maybe call- 
ed d, so that if x represent that portion of i which 
determines the point or particular phase in any in- 
terval at which any one ray of Ro leaves the crys- 
tal, x + d will be that for the corresponding ray of 
R e. The same may be said of L o and L e. 
Moreover, the rays of either of these two pairs of 
pencils are oppositely polarized, viz: in planes in- 
clined + 45° and — 45° to the plane of primitive 
polarization, and all the circumstances of their 
phase and position may be thus stated : 
Ray Phase Inclination of Plane of polr. to P. P. 
R o x + 45°. 
Re X + d —45°. 
L o x +45 
he x + d — 45. 216 
INTERFERENCE OF POLARIZED LIGHT. 
Calling 8, the difference of phase of the ordi- 
nary and extraordinary rays, into which each of 
the above pencils is divided by the Iceland spar, 
the statement becomes, 
A 
Rays Phase Incl. plane. 
R o o x 0° 
R eo x -f & 0° 
L oo x 9° 
Leo x + d 9° 
B 
Rays Phase Incl. plane,&c. 
Roe x + <5 90°. 
Ree X-\-d-\-8 90°. 
L o e x -f- 90°. 
Lee x -f- d-\- 8 90°. 
If the rhomb of Iceland spar be of considera- 
ble thickness, the group A will be so far removed 
from the group B as to prevent the possibility of 
their mixing with each other ; the rhomb is, in the 
first place, so taken. 
Considering the first group A, R o o may com- 
bine with L oo, and since they have been equally 
accelerated, having no difference of phase, they 
should, if they interfere, produce fringes immedi- 
diately about the axis plane; and the same is true 
of Re o and Leo, and hence this set of fringes 
should be of double intensity. 
Next Roo and Reo, may combine, but being 
differently accelerated, their fringes would be 
thrown to one side. Reo and Roo, may also 
combine and produce fringes as far removed to 
the opposite side of the axis plane. INTERFERENCE OF POLARIZED LIGHT. 
217 
Thus in the ordinary group A, there should be 
three sets of fringes, and the same number in the 
extraordinary group B for the same reason. Now 
these fringes are actually found to exist as may be 
seen by viewing the arrangement above described 
through a lens. It is evident that the lateral sets of 
fringes are formed by the interference of those 
rays that had been oppositely polarized by the sul- 
phate of lime, and afterwards reduced to the same 
plane, and hence the truth of the 4th law. 
If now, the Iceland spar be diminished in thick- 
ness till the group B be superposed upon the 
group A, we should expect to find but three in- 
stead of six sets of fringes, the middle one being 
still the brightest. But the fact is, but one set is seen, 
the lateral fringes disappearing entirely ; which 
shows, that the colors resulting from the inter- 
ference of the ordinary rays refracted by the 
rhomboid are complimentary to those produced by 
the interference of the extraordinary rays, and, 
therefore, that half an interval or space, 
to produce this effect, must be gained or lost when 
a ray passes from the ordinary to the extraordi- 
nary class. 
131. In the case of a diverging pencil of natu- 
ral light transmitted along the axis of a doubly 
refracting crystal, with which wTe commenced this 
subject, the rays are primitively polarized in INTERFERENCE OF POLARIZED LIGHT. 
planes at right angles to each other, and hence 
law 2d, the absence of the rings. 
Besides, the axis of the pencil being coincident 
with that of double refraction, every ray, both or- 
dinary and extraordinary, will lie in a plane of 
principal section ; the planes of polarization of the 
former will, therefore, radiate in all directions from 
the axis, while those of the latter will be perpen- 
dicular to these, and may be regarded as so many 
tangent planes to the cones formed by rays at the 
same distance from the axis. The analyzer A 
(fig. 97), being presented under the polarizing 
angle to the emergent pencil, all the ordinary rays 
in the plane of principal section which is coinci- 
dent with that of reflection D C will be reflected, 
but the extraordinary rays of this plane will dis- 
appear ; and all the ordinary rays in the princi- 
pal section A B, at right angles to this, will 
disappear, while its extraordinary rays will be 
totally reflected, from the principal sections inter- 
mediate between these, there will be partial reflec- 
tion of both classes of rays, the number of each 
being equal at the azimuth of 45° ; the ordinary 
rays being in excess when the azimuth is less than 
this, the extraordinary when greater. By reflec- 
tion from the analyzer the planes of polarization 
are brought together, and the reflected pencil will 
be polarized in the same plane; but its rays hav- 
ing been originally polarized in ojiposite planes, 
there will still, according to law 3d, be a defalca- 
tion of the rings. INTERFERENCE OF POLARIZED LIGHT. 
219 
If instead of a pencil of natural light, one 
wholly polarized in the same plane be transmitted 
through the plate D (fig. 98), and the analyzer be 
applied under the polarizing angle, and in such 
position that the plane of incidence make an angle 
of 90° with that of polarization, the rings, exhibit- 
ing the most beautiful colors intersected by a black 
cross (fig. 99), will appear. If the crystal re- 
main stationary and the analyzer be turned about 
the axis of the polarized pencil, the black cross 
will appear to dilate and grow fainter till the plane 
of incidence has been turned through an azimuth 
of 45°, beyond which, the cross, continuing to in- 
crease in brightness, will contract, and, finally, 
when the azimuth becomes 90°, or the plane of 
primitive polarization coincides with that of inci- 
dence, obtain its former dimensions and be per- 
fectly white as in figure 100. Continuing the 
motion of the analyzer, the reverse appearance will 
take place and the black cross be restored when 
the azimuth is equal to 180°. 
The explanation of all this seems quite obvious. 
The original pencil being wholly polarized in one 
plane, those of its rays which pass through the 
plate in the principal section which coincides with 
this plane, will all undergo ordinary, while those 
contained in the principal section at right angles 
to this, will undergo extraordinary refraction, with- 
out having their planes of polarization in the least 
changed. In the first position of the analyzer not 
one of these rays will be reflected, and hence the 220 
INTERFERENCE OF POLARIZED LIGHT. 
black cross. When the plane of reflection is 
brought, by turning the analyzer, to coincide with 
the plane of polarization of these rays, which is 
that of the original pencil, they will all be reflected, 
and hence the white cross. The rays that pass 
through in principal sections other than those 
mentioned, will be doubly refracted agreeably to 
the law expressed in equation (103), « represent- 
ing the angle made by any principal section with the 
plane of primitive polarization, and each princi- 
pal section will contain a set of ordinary and ex- 
traordinary rays whose planes of polarization will 
be at right angles to each other ; these falling upon 
the analyzer, in its first position, with their planes 
inclined to that of reflection between the limits of 
total suppression and reflection, will undergo par- 
tial reflection, and again be reduced to the same 
plane ; and because one set, in this case the ordi- 
nary, was accelerated more than the other, and, 
having once belonged to a pencil or beam wholly 
polarized in the same plane, they will, according 
to law 4th, interfere and produce the fringes. 
For the reflecting analyzer, substitute one of 
double refraction, say of Iceland spar, with its 
plane of principal section at right angles to that of 
primitive polarization. Two sets of fringes will now 
be seen, each exhibiting colors complimentry to the 
other, the ordinary image showing the black cross 
and extraordinary image the white. 
The crosses are, as before, easily accounted 
for. In the position assumed for the analyzer, all INTERFERENCE OF POLARIZED LIGHT. 
221 
rays not divided by the first crystal, being those 
contained in two planes perpendicular to each 
other, one coinciding with the primitive plane of 
polarization, will undergo extraordinary refraction, 
and will hence be seen in the extraordinary, but 
not in the ordinary image. 
To explain the formation of the colored rings, 
assume any ray of the original beam, say one 
contained in a principal section M N (fig. 101), in- 
clined 45° to the plane of primitive polarization ; 
and let A represent the system of semi-inter- 
vals of this ray before entering the crystal M N. 
In passing this crystal, the ray will be divided 
l 
into two equal portions, and d + ~6will, agreea- 
bly to 5th law, be the difference of phase with which 
the rays will emerge, and B will represent the 
semi-intervals after the first refraction ; the rays 
will also, on account of the small thickness of the 
crystal, be superposed, or nearly so, and be in a 
condition to interfere if their planes of polariza- 
tion were the same. On reaching the analyzer, 
each ray is again divided into two equal parts, the 
ordinary rays of each preserving, with reference to 
each other, the same phase C, as at B ; but the 
extraordinary portion of the first ordinary ray, 
gaining or losing by law 5th, half an interval on 
the extraordinary portion of the first extraordinary 
ray, these rays will have the analyzer in the con- 
dition represented at C'. The rays of each class 
passing the analyzer parallel, will emerge in the 222 
CIRCULAR POLARIZATION. 
same state, be superposed, and produce, by their 
interference, complementary colors. 
Circular Polarization. 
As a general fact, it has been observed, that 
when a polarized ray is passed through a doubly 
refracting crystal in the direction of its optical 
axis, it undergoes no change, the analyzer pro- 
ducing the same effect after as before transmis- 
sion on being offered to the same side of the ray. 
To this, however, there are some remarkable 
exceptions, one of the most conspicuous of which 
is exhibited by rock crystal, or quartz. When 
the analyzer is in the position affording no reflec- 
tion, the interposition of a crystal of this substance 
will restore a portion of light, and to cause it to 
disappear, the analyzer must be turned through a 
certain angle about the ray, the magnitude of this 
angle depending upon the thickness of the inter- 
posed quartz. As the thickness of the mineral is 
increased, the rotation of the analyzer must be 
continued. The plane of polarization is thus 
twisted into a surface of double curvature re- 
sembling the turns of an auger, or the surface 
generated by the rotation of one right line about 
another perpendicular to it, at the same time that 
it has motion of translation along this second line. 
This is called circular polarization. 
It has been found that for a given thickness, the CIRCULAR POLARIZATION. 
223 
arc of rotation necessary to bring the analyzer 
into the position of evanescence is different for the 
differently colored rays, this arc being given by the 
formula 
k t 
r “ "V ' 
in which r represents the arc, k a constant, t the 
thickness, and i the interval for the particular 
color. 
It has also been ascertained, that for some speci- 
mens of quartz, it was necessary to turn the ana- 
lyzer in one direction, while for others, in an op- 
posite direction, and that a singular connection ex- 
ists between this property and the right or left 
handed direction in which certain small faces of 
the crystal lean around the summit of the variety 
called plagiedral quartz. 
Other bodies, besides quartz, possess the pro- 
perty of circular polarization, but in different de- 
grees ; and if two of these bodies be interposed, 
the arc of rotation is that due to the sum or dif- 
ference of their thickness, according as they pos- 
sess the same or opposite kinds of circular polar- 
ization. Or more generally, 
r T = r t + r t-f- r" t" -{- &cc., 224 
ELLIPTICAL POLARIZATION 
in which R is the rotation due to the combination; 
T its entire thickness; r, r'f Sec., and t,t',&L c., 
the corresponding quantities answering to the 
several individuals of the combination ; the pro- 
ducts entering the expression with the same or 
different signs, according as the different media 
tend to turn the plane of polarization in the same 
or different directions. This formula is found to 
hold good not only with solid crystals, but also 
with liquids, possessing the property of circular 
polarization, when mixed together. 
Elliptical Polarization. 
Where light is polarized by reflection from the' 
surfaces of transparent media, the entire beam 
has but one plane of polarization, and that in the 
plane of reflection. But when metallic surfaces 
are employed, the beam is found to consist of two 
portions of unequal intensity polarized in op- 
posite planes. These portions are superposed,, 
and the intensity of that polarized in the plane 
of reflection is always the greater ; its excess, ac- 
cording to Sir David Brewster, above that polar- 
ized in the opposite plane, varying with the dif- 
ferent metals, being least in silver and greatest in 
galena. COLORS OF THIN PLATES. 
225 
Colors of Thin Plates, 
Transparent media, when reduced to very thin 
films, are found to exhibit colors which vary with 
the thickness of the film. These are called the 
colors of thin plates, and the easiest way to ex- 
hibit them is by means of a soap bubble blown 
from the end of a quill or the bowl of a common 
smoking pipe. As the bubble increases in diame- 
ter, and the fluid enveloped is reduced in thickness 
at the top by gradual subsidence toward the bot- 
tom, many colored and concentric rings will be 
seen around the point of least thickness. At this 
point, the color will be found to change, first ap- 
pearing white, then passing through blue to per- 
fect blackness, the rings the while dilating till the 
bubble is destroyed. 
The same is true of any other medium, whether 
gaseous, fluid or solid. 
These different colors being exhibited upon the 
same plate of variable thickness, no single color 
can be identified with its chemical composition. 
When of uniform thickness, a single color only 
will be seen, and this will change as the thick- 
ness of the plate changes. 
A thin plate is very conveniently formed of air; 
and for this purpose, let A B (fig. 102), be a plano- 
convex, and C D a plano-concave lens, placed one 
upon the other, as represented in the figure. 
When this arrangement is viewed on either of the 226 
COLORS OF THIN PLATES. 
plane faces by reflected light, colors will be seen 
in the form of concentric circles about the point 
of contact, which, should the pressure be sufficient, 
will be totally black. If viewed by transmitted 
light, rings complementary to the former will ap- 
pear about the central spot which will now be per- 
fectly white. In homogeneous transmitted light, the 
rings are alternately bright and dark, beginning 
with the central spot; and by reflected light, dark 
and bright. They are broadest and have the 
greatest diameter in the red, and narrowest with 
least diameter in the violet; the breadths and 
diameters in the other colors being intermediate 
and varying in magnitude in the order of the 
spectrum from red to violet. It is by the super- 
position of these rings that the different colors ap- 
pear in common light. 
These colors, which are of different orders as 
regards tint, constitute what is called Newton’s 
scale ; and by reflected light, occur as follows, 
beginning with the central spot. 
ls£ order. Black, very faint blue, brilliant 
, white, yellow, orange and red. 
2d order. Dark violet, blue, yellow-green, bright 
yellow, crimson and red. 
3d order. Purple, blue, rich green, fine yellow, 
pink and crimson. 
4th order. Dull blue-green, pale yellow-pink, 
and red. 
5th order. Pale blue-green, white and pink. 
§th order. Pale blue-green, pale pink. COLORS OF THIN PLATES. 
227 
1th order. The same as 6th, very faint. The 
other orders being too faint to be 
distinguishable. 
If the plano-concave lens be replaced by a me- 
tallic reflector, and the light be polarized in a plane 
at right angles to that of incidence, the rings will 
disappear when the rays are incident under the 
maximum polarizing angle. Under these circum- 
stances the light will be wholly transmitted at the 
first surface, but will be reflected of considerable 
intensity at the second ; hence it is inferred that 
the first surface is essential to the formation of the 
rings. When the metallic reflector is slightly tar- 
nished, a second system of rings has been ob- 
served, arising, as is supposed, from the light irre- 
gularly reflected in consequence of the tarnish; 
this proves the agency of the second surface in 
ducing the rings, which are in consequence attri- 
buted to the interference of the rays reflected 
at the first surface with those reflected at the 
/ 
second. 
Suppose a small beam incident perpendicularly or 
nearly so, on the first surface of the plate, where 
the thickness is t. A part will be reflected back 
along the incident beam, the rest, being transmit- 
ted, will traverse the thickness t. At the second 
surface a part is again reflected, and the reflec- 
ted portion returning through the thickness t, will 
emerge at the first surface in the direction of the 
incident beam, and be superposed on that re- 
flected at this surface. This superposition may COLORS OF THIN PLATES. 
take place according to any of the conditions 
explained in (105), with respect to the inter- 
vals i. If dissimilar points are superposed, dark- 
ness will result, and to this end the difference of 
route traversed, which is 2 t, must be equal to 
i 
or some odd multiple of it. 
But the portion reflected at the second surface 
will, in part, be again reflected at the first, and will 
traverse the thickness t a third time, and emerge 
below superposed upon the portion first transmit- 
ted at the second surface. The difference of 
route of these portions will also be 2 t, so that 
they should also give total darkness. Experiment 
shows, however, that this is not the case, for where- 
ever there is total darkness by reflection, there is a 
maximum of brightness by transmission. Hence, 
if interference be the cause of the rings, there 
must be half an interval added to or subtracted 
from the route at each internal reflection. This 
will give for the interfering rays, in case of reflec- 
ted rings, a difference of route expressed by 
i 
2<+t; 
and for the transmitted, 
2 t + K 
To ascertain the value of t, at the different COLORS OF THIN PLATES. 
rings, call d the diameter of any one of them, as 
determined by actual measurement, r and r the 
radii of the surfaces, v and v', the versed sine of 
the arcs whose sines are equal to the semi-diameter 
of the ring in question. 
Then, for very small arcs we have, (fig. 102), 
(4): 
V = 1 
2 r 
and 
2 r ’ 
whence 
rf2/1 1\ 
In this way Newton found the thickness at the 
brightest part of the first ring after the central black 
spot, 0,00000561 of an inch. He also found the 
diameters of the darkest rings to be as the square 
roots of the even numbers Q.2.4.6.&C., and 
those of the brightest as the square roots of the 
odd numbers 1.3.5.7. &c. The radii of the sur- 
faces being great compared with the diameters of 
the rings, the value of t at the alternate points of COLORS OF THIN PLATES. 
greatest obscurity and illumination are as the natu- 
ral numbers 
0 . 1 . 2.3.4 . &c.f 
hence, the value of t, just found, multiplied by these 
numbers will give the thickness at the different 
rings. 
On comparing the value for the thickness at the 
first bright ring, with the numbers in the table of 
article (106), it will be found just equal to one 
fourth of the interval denoted by l, for the yellow 
ray, which is the most illuminating of the elements 
of white light. 
Taking this value for t, we shall have for the 
difference of route of the interfering rays pro- 
ducing the dark rings by reflection, including the 
central black spot, 
l 31 51 7i p 
21 2 9 2 7 2 7 
these being the even multiples of -j, increased 
by ~2> for the acceleration or retardation by one 
,v.., .• > . • 
internal reflection. 
The odd multiples give 
t, 2tt 31, &c., &c., COLORS OF THIN PLATES. 
231 
from which it is obvious, that the transmitted rays 
will be complementary to those seen by reflection* 
The phenomena we have just considered are 
equally produced, whatever may be the medium 
interposed between the glasses, the only differ- 
ence being in the contraction or expansion of the 
rings depending upon the refractive power of the 
medium. It is found that as the refractive power 
of the medium increases, the diameter of the rings 
will decrease, which might have been inferred 
from article (107). 
THE END.