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Francis A. (Francis Alexander) Tarleton.

# An introduction to the mathematical theory of attraction

. (page 10 of 11)

the first and last of equations (49), we get

/sin 2i sin 6 = I { {sin (i + 1\) cos (* - t'j) sin t + sin 2 1\ tan Y_, )

(50)

In like manner, from the second and third we obtain

/ sin 2 cos = /i sin (i + i,) cos 0^
Hence, by division, we get

tan = cos (i - f,) tan Oi + . " <! . . tan Y,. (51)
sin (t -f ,)

The second value of tan 6 is obtained by putting 2 and /,
for 0| and ii in (51).

274. L nia\al Crystals. In the case of what are called
uniaxal crystals, Fresnel's ellipsoid is a surface of revolution.
If we suppose c = b in the equation of the wave-surface (29),
Art. 261, that equation becomes

that is, (f*-P)(aV+#

But (52) is the equation of the surface composed of
sphere whose equation is r" = i*, and the ellipsoid of revo-
lution whose semi-axis of revolution is b, and whoso other
semi-axis is a.

If a > b, and Fresnel's ellipsoid is prolate, the ellipsoid
forming part of the wave-surface is oblate.

ITniaxal Crystals. jyg

These conditions hold good in the case of a crystal of
calcium Carbonate commonly called Iceland spar This
crystal is very celebrated in the history of science "s
observations of its behaviour led to the discovery of double
refraction and of polarized light.

The axis of revolution of Fresnel's ellipsoid is coincident
with the line which is called the axis of the crystal This
me is the axis of symmetry, and can be determined from
the geometrical form of the crystal.

In the case of an uniaxal crystal, all rays inside the
crystal whose directions of electric displacement are perpen-
dicular to the axis are propagated with the same velocity
Ilm appears from (25) by making C = B and A = then'
M 2 + v = 1, and V 2 = B\ Conversely, if

we have (A 2 - 2 )cos 2 S = 0, and therefore S = ~. For
these rays the wave-surface is a sphere.

Again, if the wave-surface be an ellipsoid of revolution
since the normal to a surface of revolution meets the axi'
the ray, the wave-normal, and the axis must be in the same
plane ; but the plane containing the ray and the wave-normal
by Art. 260, contains the direction of electric displacement'
Hence this direction is in the plane containing the ray and
the axis.

When light passing through an isotropic medium is
refracted at the surface of an uniaxal crystal, one refracted
ray is refracted in the same manner as "if the crystal were
isotropic, since the wave-surface of this ray is a sphere.
I his ray is called, therefore, the ordinary ray. The other
refracted ray, whose wave-surface is an ellipsoid of revolution,
is called the extraordinary ray.

Both rays are polarized, and as a result of experiment it
is said that the ordinary ray is polarized in the principal
plane. By the principal plane is meant the plane passing
through the refracted ray and the axis of the crystal. Hence
we see again that the direction of electric displacement is
perpendicular to the plane of polarization.

N2

180 Elccti-uiiKKjm-lic Theory of L'ujht.

275. Uniaxal Crystal. Reflexion and Itefrac-
tlon. In the case of an uniaxal costal, since \\ = 0,
equutioiis (49) beeomu

sin cos i (I sin 9-1' sin ff) = J, sin i\ cos n sin 6^

+ It (siu f a cos i 2 siu0 2 + Biii*/ 8 tan x*)>
sin /(7cos0 + 7'cos0') = I\ sin?', cos0i + / 2 sinfjcos
cos /(/cos0 - 1' cos 61') = /i cos?'i cos0i + 7 2 cos/ 2 cos 9 2 ,
I em 6 + I' sin 0' = J, sin 9 t + 7 2 sin 2 .

As an example of the use of these equations, we may
suppose light to fall on the surface of an uuiaxal crystal cut
perpendicular to the axis.

In this case, since the axis of the crystal is the normal to
the surface, the plane of incidence contains the axis and the
wave-normal of the extraordinary ray, and, consequently, the
ray itself. Hence both refracted rays are in the plane of
incidence ; and, by Art. 274, we have

Accordingly, equations (53) become

sin i cos i(I sin 9-1' sin 6") \

= / 2 (sin ? 2 cos it + siiiV 2 tan ^ 2 ),

sin i(Icioad + /' cosfl'j = 7, sin i,, v (54)

cos (7cos0 - 7' cos0') = Ii cos t\,
7 siu + I' siu 0' = 7 2 .

If we now suppose that the incident light is polarized in
the plane of incidence, = 0; and from the first and last of
equations (54) we have

It (sin i cos i -f sin i t cos i, -f siiiV 2 tan \,) = 0.

Un'mial Crystal. 181

Since the expression by which 7 2 is multiplied cannot be
zero, we get 7 2 = 0, and therefore sin 9' = 0. The second
and third of equations (54) become, then,

sin / (7 + /') = 7, sin ? cos i (I- I'} = I, cos t\
whence we get

27 sin / cos i = ^ sin (t\ + i), 21' sin i cos i = sin (i\ - i).
Finally, we obtain

L=I sin2 ' r _ r si" ('. - i)

sin (*+!,)' sTnTTwo'

Again, if the incident light be polarized in a plane
perpendicular to the plane of incidence, 9 = ?, and the
second and third of equations (54) become

l e sin i cos 9' = 7 X sin /\, - /' cos i cos Q' = l^ cos i\.
Hence we obtain 7, sin (i + i\) = 0, and therefore ^ = ;
whence also cos 9' = 0, and 9' = -

From the first and last of equations (54} we have, then,
sin i cos i (I- 1'} = 7 2 'sin i 2 cos H + sin 2 / 3 tan ^2), I + I' = 7 2 .
"Whence

,, _ j s i n * cos z ~ ( s i n 4 cos 4 + sin 2 / 2 tan ^ 2
sin i cos a" + sin i\ cos ?' 2 + sin 2 2 tan ^ 2

r _ r 2 sin z cos

sin e cos + sin ?' 2 cos ? 2 + sin 2 2 tan ^ 3 '

These expressions can be put into a simpler form.

The angle x is the angle between the directions of electro-
motive force and electric displacement, and is measured from
the former towards the latter in the same direction as the
line of displacement is turned in order to become the wave-
normal. This appears from the figures and formulae of

182

Electromagnetic Theory of Liy/tt.

Arts. 260 and 272. What has been said amounts to this
that in equations (53) ^ is to be regarded as positive when
the direction of the electromotive force does not lie between
those of the displacement and the wave-normal, and conse-
quently the ray does occupy this position. In the present
case the axis-minor of the wave-ellipse is the normal to the
surface, and the positive angular direction is from it to the
refracted wave-normal. The refracted ray lies farther from
the axis than the normal, and consequently does not lie
between the electric displacement and the wave-normal.
Hence in (56; the angle %? is negative.

An expression for tan x can be found by the geometry of
the ellipse.

FIG. 5.

In the figure OX represents the axis of the crystal, OQ the
line of electromotive force, OP that of electric displacement,
01 the extraordinary wave-normal. Then v is the angle
QOP ; but QOP = R01, and

= UI ' OI = 2 triangle ROI
01 = OP 01* ~

Now, if pi and jh be the focal perpendiculars on the
tangent, and ti and U the intercepts on the tangent between

their feet and the point of contact, = , and therefore

(p<+p,)(t,-t,) - (/>, -

Ui/id.i'fil Crt/sfa/.

183

but (;;, + jh)((\ - tz) is double the area of the triangle
ROI, and | Q0i -Jfc)(& + &) is double the area of the
right-angled triangle whose sides are ^/((i 2 - b z ) sin t' 2 and
v/(rt 2 - I'} cose'z, the angle XOI being ,. Hence, if Of be
denoted by ;j, we have

In the present case ^ 2 is negative, and we have

(a 2 - i*) sin 2 / 2 \
sin e, cos it + sin- / 2 tan v 2 = sin ? 2 cos Ml-

\ fl 2 sin 2 / 2 + 6 2 cos 2 ? 2 y

J 2 sin i-i cos / 2 jB 2 sin 2 cos ' 2

a z sin 2 ? 2 + i 2 eos 2 it A 2 sin 2 / 2 + B* cos 2 / 2

Again, if Fdenote the velocity of propagation in the external
medium,

F 2 sin 2 / 2 = (^4 2 sin 2 ? 2 + ^ 2 cos 2 ^ 2 ) sin 2 / ;
whence

sin 2 h + B 2 cos 2 / 2

F 2 -

Hence

B 2 sin / 2 cos / 2 5 sin t

^'Bin'fc + .B'oos 8 /,

and

sin/ cost + sin / 2 cos/ 2 + siu 3 '/ 2 tan ^ 2

sin e ( F 2 cos i + B </ F 2 - A 2 sin 2 /j
V i

siiu'cos? - (sin/ 2 cos/ 2 + si

sin? { F 2 cost - B </ F 2 - A 2 sin 2 ?!
F 2

(58)

184 Electromagnetic Tli<or>/ of Liyht.

Accordingly,

F'oos/- 7? y/CF 2 - ^4' si n 2 ?)
F 2 cos > + v/( F' - ^4* wn/) '

2 F 8 cos i

7, = 7

F 8 cos i + 7V( V* - A* si

If the value of i be such that I' - 0, the reflected ray,
when common light falls on the crystal, is polarized in the
plane of incidence. This value of i is called the polari/ing
angle of the crystal when cut perpendicular to its axis.

Making 7' = in (59), we have

F 4 (l - sin'i) - &( V* - A* sin'*) ;

whence

F ; (F 8 -7J 2 )

F* -^ 8 /y 2

(60)

276. Reflexion and Refraction at Interior Surface

of Crystal. When light passes from the interior of an
uuiaxal crystal into an isotropic medium, there are, in gem- ral,
two reflected rays ; and when the incident ray is nn ordinary
ray, we have

sin /i cos f\ (/, sin Oi - 1\ sin
+ (sin i\ cos i\ sin W*

tan \ z ) l

= 7 3 sin ? 3 co8/ s sin 0.,
sin f\(7i cos Oi + I\ cos 0',) + 7' a sin i\ cos 0' 2 ,

= 7 3 sin /s cos ^ 3 ,
cos ;',(/i cos 0i - 7'i cos 0'i) + 7'j cos ?" 2 cos 0' a .

= 7 3 cos / 3 cos 3 ,
7, sin 0! 4 /', sin 0\ + 7', sin 0' a = 7, sin 8 .

^

Reflexion and Refraction. 185

When the incident ray is an extraordinary ray, the
equations at the refracting surface become

(sin it cos / 2 sin 2 + sin 2 ? 2 tan ^ 2 ) J 2 1

+ I' \ sin i\ cos i'\ sin O' t \
+ (sin /' 2 cos i'z sin 0' 2 + sin 2 ' 2 tan x' 2 ) 7' 2

= / 3 sin 4 cosz'g sin 0,3,

/ 2 sin / 2 cos 2 + I\ sin ?'i cos 0\ + 7' 2 sin A cos 0' z <. (62)

s 7 3 sin / 3 cos
/ 2 cos / 2 cos 02 + I' i cos A cos B'i + I' z cos A cos B'z

/ 3 C(^S ?3 COS

/ 2 sin 2 + l'\ sin 0'j + /' 2 sin r 2 = / 3 sin 3 . j

When the crystal is cut perpendicularly to its axis,

In this case, the first and last of equations (61) become
(sin <" 2 cos i' z + siir/' 2 tan ^' 2 ) 7' 2 = / :3 sin 3 sin ?' 3 cos /j,

/'a = / 3 sin 03.
Hence

/ 3 sin 03 {sin / 3 cos ^ - (sin A cos i'z + sin 2 A tan x'z)} = ;

but. the multiplier of 7 3 sin 3 in this equation is nor, in
general, zero, and therefore we have

sin 03 = 0, I', = 0.

Consequently there is no extraordinary reflected ray, and the
refracted ray is polarized in the plane of incidence. From the
second and third of equations (61) we then obtain

sin (>, - i.) si 'i_ / 63)

186 Electromagnetic Theory of Light.

If the ray incident on the interior surface of the crystal
bo the extraordinary ray, and the crystal, as before, he cut
perpendicularly to its axis, since all the rays and wave-
normals are in the plane of incidence which cuts the wave-
ellipsoid in an ellipse whose axis-minor is the axis of the
crystal and also the normal to the surface, we have

i'a = TT - i,, x's = - X

whence

sin i't cos i't + sinYa tan x' a = - (sin it cos / 2 + sin 2 / 2 tan xJ-

In this case, the second and third of equations (62) become

/' 'i sin /'i = I s sin / 3 cos #,, I'\ cos ?', = I 3 cos / 3 cos 3 ,

whence I 3 cos 3 sin (/ 3 - i'i) = ; but siu(i 3 -i'i) cannot be
zero, and therefore cos0 3 = 0, and 7', = 0. Consequently,
there is no ordinary reflected ray, and the refracted ray is
polarized in a plane perpendicular to the plane of incidence.
The first and last of equations (62) now become

(sin it cos it + sin 2 it tan x 8 ) ( ?t ~ 1'*) - fa g i u ' cos ''>

It + I't = /a-
Hence

sin i 2 cos it + sin 2 / 2 tan x^ - sin / 3 cos / 3
2 sin it cos it + sin 2 / 2 tan x a + g i n '3 cos / 3 '

2 (sin it cos it + sin'/a tan x)

sin it cos it + smV 2 tan x a + sin h cos 3

By reductions similar to those effected in the ca-e of
-equations (56) we get.

, 2 - A* sin*/,) + F 3 2 cos /, '

a 2 - A 1 sin 2 / 3 ) - Fs" cos / 3

-r j & L* *y y r a ** Dill 13;

2 ' /T ^ 3 2 -^ 2 8 in 2 / 3 )-f F 8 2 oos/ 3 '

Singularities of the Wave-Surface. 187

If an uniaxal crystal, bounded by faces parallel to each
other and perpendicular to the axis, be placed in an isotropic
medium and a ray of light polarized in a plane perpendicular
to the plane of incidence be transmitted through the crystal
the incident and emergent rays are parallel, and the plane of
polarization remains unchanged. Then ,', = ,', and in virtue
of equations (59) equations (64) become

(65)

When the incident ray falls on the first surface of the
crystal at the polarizing angle, we have

l' = 0, 7' 2 = 0, and J 3 = 7 2 = /.

In this case, the incident light passes through the crystal
unchanged in intensity, direction of electric displacement,
and direction of propagation,

277. Singularities of the Wave-Surface The

equation of the wave-surface, Art. 261, may be put in
the form

<V 4 by + cV - V)0*; S + if + s 2 - V) - (<? - V)(b* - c 2 )// 2 = 0.

From this equation it appears that if the point of intersection
of the three surfaces

V + I? if + cV - rte = 0, x 2 + i' + z* - b 2 = 0, // =

be taken as origin, the lowest terms in the equation of the
wave-surface are of the second degree, and therefore that the
origin is a double point on the wave-surface at which there
is a tangent cone of the second degree.

188 Electromagnetic Throry of Light,

If we seek for the coordinates of the points of intersection
of the three surfaces, we have

V + cV = rtV, x* + z> = b\ >/ = ;

whence we obtain for the coordinates of the point the

expressions

,.j M ;, _ r t

*-'\$< **-> *' = a ^- (fi6)

The equation of the circular sections of Fresnel's ellipsoid

whence, if w,, ts tt and za 3 denote the direction-cosines of the
perpendicular to a plane of circular section, we have

, fi ~

From (66) and (6?) it appears that a singular point on
the wave-surface is on a perpendicular to the plane of &
circular section of Fresnel's ellipsoid at a distance l> from
the origin.

The existence of such points follows readily from the mode
of generation of the wave-surface described in Art. 260. Fiona
thence it appears that the perpendicular to each section of
Fresjiel's ellipsoid meets the wave-surface in two points whose
distances from the centre are equal to the principal semi-axea
of the section.

If the section be circular, every axis is a principal axis,
and all the corresponding points on the wave-surface coalesce
into one.

The perpendiculars on the corresponding tangent-planes
of the ellipsoid are, however, not in the same plane ; and thus
corresponding to the one ray going from the centre to the
singular point there are an infinite number of wave-front*
that is, an infinite number of tangent-planes to the wave-
surface meeting at the singular point.

Singularities of the Wave- Surf nee. 189

As the wave-normals and velocities of propagation are
different for these fronts, when the ray reaches the surface
of the crystal it is refracted into an infinite number of rays,
forming a cone, and the phenomenon exhibited is termed
conical refraction.

From the consideration of the ellipsoid reciprocal to
Fresnel's ellipsoid, it is easy to see that the wave-surface
must possess singularities of another kind in addition to
those mentioned above.

From Art. 260, it appears that the perpendicular to each
section of the reciprocal ellipsoid is perpendicular to two
tangent-planes of the wave-surface, and meets them in points
whose distances from the centre are the reciprocals of the
semi-axes of the section. If the section be a circular section,
every axis is a principal axis, and all the corresponding feet
of perpendiculars on tangent-planes to the wave-surface
coalesce into one.

The central radii of the reciprocal ellipsoid are co-direc-
tional with perpendiculars on tangent-planes of Fresnel's
ellipsoid, which are the reciprocals of the radii, so that all
the perpendiculars to tangent-planes of Fresnel's ellipsoid
which lie in a circular section of the reciprocal ellipsoid are
equal to the mean semi-axis of Fresnel's ellipsoid, and corre-
spond to a single tangent-plane to the wave-surface. The
corresponding radii of Fresnel's ellipsoid do not, however, lie
in the same plane, and are not equal, so that there are an
infinite number of rays corresponding to the same wave-front
which must therefore touch the wave-surface all along a curve.
To find the nature of this curve, we may proceed thus.

Let p denote the length of the central perpendicular on a
tangent-plane of Fresnel's ellipsoid, and a, /3, 7 its direction-
.angles.

If /; lie in the circular section of the reciprocal ellipsoid,
we have p = b, and therefore

<i z cos 2 a + b~ cos 2 /3 + c 2 cos 2 7 = 2 (cos 2 a + cos 2 /3 + cos 2 ^) ;
that is, ( 2 -& 2 ) C08 2 a - (6 2 -C 2 ) C08 2 y = 0.

Also, cos 2 a + cos 2 7 = siu 2 /3 ;

b" 1 c 2 a 2 b~

whence cos 2 a = -, siu 2 /3, cos 2 7 = 2 _ ^ sin'/j (68)

190 Electromagnetic Theory of Light.

Let r denote the central radius of Fresnel's ellipsoid to tlie
point of contact of the tangent-plane perpendicular to j), then

(i* cos 2 a + i 4 cos 2 /3 + c* co.-r-y
~V~

and if p denotes the distance of this point of contact from
the foot of the perpendicular, /o 8 = r 9 - ;A In the present
case, p = b, and we have

a* cos 2 n + b* cos 2 /3 + <* cos 2 y - 4*

p* - j-r-

_ (r 4 - &') cos 2 - (6* - /') cos ? Y
P

Substituting for cos 2 a and cos 2 7 their values from (68), we get

It is plain, from the construction in Art. 260, that p is
the distance from the foot of the perpendicular on the tangent-
plane to the wave-surface to its point of contact, and that this
distance is parallel to the corresponding direction of displace-
ment in the wave-plane. In the present case the wave-plan!
contains the axis of y, and /3 is the angle which the electric
displacement makes with this axis. Hence j3 is the angle
which the line from the foot of the perpendicular to the
point of contact of the wave-front with the wave-.surt;i( .-
makes with a parallel to the axis of y in the wuve-front.

Accordingly,

-*>*. ft (69,

is the equation of the curve along which the wave-front
touches the wave-surface. This curve is therefore a circle
which touches the parallel to the axis of y at the foot of the
perpendicular from the centre, and whose diameter is denoted
by the expression

Total Reflexion. 19j

Corresponding to the wave-plane we have been considering
there are an infinite number of rays which meet the wave-
iront along its circle of contact with the wave-surface. All
these rays have the same wave-normal, and are propagated
with the same normal velocity. Hence, when they are re-
fracted at the surface of the crystal, the emergent rays are
parallel and form a cylinder. Unless the wave-normal be
normal to the surface, the section of this cylinder made by
the plane bounding the crystal is an ellipse.

The remarkable phenomena described above were foretold
by Hamilton as consequences of properties of Fresnel's wave-
surface discovered by him. They were realized experimentally
nrst by Lloyd, and long afterwards by Fitzgerald.

278. Total Reflexion. When light passes from a denser
into a rarer medium, if the angle of incidence exceed sin" 1 -,

where ^ denotes the relative index of refraction of the medm
there is no refracted ray. In fact, under these circumstances^
a refracted wave-plane is impossible, as it would in the case
of an isotropic medium, be a tangent-plane to a sphere drawn
through a line lying inside the sphere. If both media be
isotropic, equations (43) seem impossible to satisfy ; for, if
we suppose /! zero, these equations cannot be satisfied unless
we make I and /' each zero.

Mathematically it is possible to give a solution of equa-
tions (43), which in its final result is physically satisfactory ;
but it seems impossible to obtain a sound physical basis for
the equations themselves.

The mathematical solution is as follows : Assume
D = ae-<4>, I)' = a'e-'p', D 1 = aj e-i>i,

where < = v/- 1 , and (/> = -^ { Vt - (Ix + my 4 ws) j,

A

'

AI

then the differential equations of wave -propagation are
satisfied, and D, &c., are periodic.

192 Electromagnetic Theory of Light.

If we now suppose the incident light polarized in the
jilane of incidence, since

Fi 1 V 1

A7 = ^' T = ^' and ri = T '

at the origin, where x, //, and x are all zero, we have

f - f' ~* * f J

and as equations (43) mathematically hold good, we have
sin 1 1 cos i - cos , sin t r
sin f , cos i + cos t\ sin

But sin ', = /z sin i, cos i x = t -v/O* 2 sia 8 ' ~ !)

and therefore

/' fi sin t cos - i sin v/(/u 2 sin 2 i - 1) 1 - t tan e
/ a sin i cos t + t sin & *S (\$ sin 2 * 1) 1 + i tun t

v/^sii. 2 /-!)
= (cos c + ( sin e s = e" 2 ", where tan e = * .

fj. COS /

Hence D' = a r 2 '* <r"J>' = a e-'(<?>'-i- 2 ),

and, accordingly, the intensity of the reflected light is equal
to tiiat of the incident ; but its phase is increased by 2e.
Again,

and since the axis of x is normal to the surface separating
the media, and the axis of z perpendicular to the plane of
incidence, we have

/j = cos i, = t </(fj? siu't - 1), nil = AI sin i, N! = 0,

2iir, ,

(70)

Absorption of Light. 193

nt r- h H S ^^f 1 ?"' ^ e .P*er of e whose index is real is
P^ 10 . d ^; and since A, is very small when * is of sensible
magnitude this factor tends to become very small Hence
at any sensible distance from the boundary A is very small
and there is no visible refracted ray.

In other cases of total reflexion a similar mode of treat-
ment may be employed The results obtained above satisfy
the mathematical conditions holding good when the reflexion
is not total, and the final result is consistent with the observed
phenomena; but the whole investigation can scarcely be
regarded as having any physical validity.

279. Absorption of tight. When a medium is not a
perfect insulator, an electromotive force produces not only a
change 01 electric displacement but also a conduction-current

i G be the electric conductibility of the medium, the
istance of an element of unit section parallel to the axis

of x is , and the electromotive force for this element is

Xdx. Hence, if i\ denote the intensity of the conduction-
current parallel to x, we have i\ = CX.

The total current is made up of the conduction-current
and that due to a change of the electric displacement;
accordingly, we have

and as X = ~f, we obtain u =/+ ^

**

ic

Substituting for u in terms of the components of magneti
force, and for the latter in terms of those of displacement, by
means of equations (13), (15), and (11), we get

f v ./ + . (7J)

K -&K dxdx d v '

The last term in this equation is zero; and if we take
the normal to the plane of the wave as the axis of z, the
displacement /is a function of z only, and (71) becomes

K

( >

194 Electromagnetic Theory of Light.

If U denote the velocity of wave-propagation when there
is no absorption, we have

and putting ir-nC = k, we get
Thus (72) becomes

To solve this equation, we may assume
/= a '0>< -*),

where /- 2?r

i = v/-l, n = ,

and m is a quantity to be determined so as to satisfy (73).
We have, then,

- n*

that is, 2

-^-4*. (74)

Assume m-q- ip, then

Eliminating g, we get
/hence

;>'' =

Absorption of Light. 195

As/> is real, p 2 must be positive, and therefore

Here A; is of the same order as C, which is of the order

K 1

or

Hence - is of the form v ^ 2 , where v is a numerical
coefficient depending on C and on the units selected, T the
time of vibration in the wave of light, and ^the unit of time.
In order that C should have any sensible magnitude, V must
be enormously great compared with r. Hence WU ' j g a

^j2

small quantity, whose square may be neglected in the
expansion of the square root, and we have

2U* 2n*
Substituting q - tp for m, we get

/ = a, ft* '(*- **L (75)

As the wave is advancing in the direction of 2 positive,
q is positive ; and since pq is positive, p must be positive.
Hence

2ir^CU, (76)

also

_ 2kn _ n _ '2Tr

q = IT ~ 1J = Ur'
and

2i7T

- (Ut- Z\

f = ae-f'eVr (77)

196 Electromagnetic Theory of Light.

It follows, from the expression obtained for f, that the
velocity of wave-propagation is U, and is therefore unaltered
by absorption. In consequence of the factor e~ ps , the amplitude
of /diminishes as z increases. Since ;; varies as C, unless C
be very small, the amplitude of /diminishes rapidly, and the
medium is practically opaque.

280. Electrostatic and Electromagnetic measure.

The reader of the foregoing pages may have been struck by
an apparent inconsistency between the present Chapter and
Chapter XI.

In Chapter XI. the specific inductive capacity k is of the
nature of a numerical quantity. In the present Chapter, the
specific inductive capacity TTis regarded as the reciprocal of
the square of a velocity. The apparent inconsistency results

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