" = ‘657, or cp" =41° 5' 30 nearly ; for air and water, m" = 1*33, and

' 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

1 + g— V7’ or «= «/'' + ' = 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

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>3 =*
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-
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. : 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 = 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 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/

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
and m are given for any particular color.
For any value of ,and consequently for ) 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 (
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 , and the first reflec-
ted beam is perpendicular to the first refracted.
Moreover,
1 1
-— = — cot cp = tan cp
m tan

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.