James Clerk Maxwell.

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part of any surface which is cut off by this cone on the side next
the positive direction of the axis is 2 ir 6 (1 â€”cos 0).

If we further suppose this surface to be bounded by its inter-
section with two planes passing through the axis, and inclined
at the angle whose arc is equal to half the radius, then the
induction through the surface so bounded is
J e (1 â€” cosd) =s *, say ;

and = cos^^ (l â€” 2 -)â€¢

If we now give to <l> a series of values 1, 2, 3... e, we shall find
a corresponding series of values of 6^ and if 6 be an integer, the
number of corresponding lines of force, including the ^is, will
be equal to e.

We have thus a method of drawing lines of force so that the
charge of any centre is indicated by the number of lines which
diverge from it, and the induction through any suiface cut off in
the way described is measured by the number of lines of force
which pass through it. The dotted straight lines on the left-
hand side of Fig. 6 represent the lines of force due to each of
two electrified points whose chaiges are 10 and â€”10 respect-
ively.

If there are two centres of force on the axis of the figure we
may draw the lines of force for each axis corresponding to values
of <t>i and <t>2, and then, by drawing lines through the consecutive
intersections of these lines for which the value of <t>i + ^^ is the
same, we may find the lines of force due to both centres, and in
the same way we may combine any two systems of lines of force
which are symmetrically situated about the same axis. The
continuous curves on the left-hand side of Fig. 6 represent the
lines of force due to the two chained points acting at once.

After the equipotential surfaces and lines of force have been
constructed by this method, the accuracy of the drawing may be
tested by observing whether the two systems of lines are every-
where orthogonal, and whether the distance between consecutive
equipotential surfaces is to the distance between consecutive lines
of force as half the mean distance from the axis is to the assumed
^unit of length*

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123.] ^ND LINES OP INDUCTION. 185

In the case of any such system of finite dimensions the line of
force whose index number ^ <t> has an asymptote which passes
through the electric centre (Art. 89 d) of the system, and is in-
clined to the axis at an angle whose cosine is 1 â€” 2 4>/e, where e
is the total electrification of the system, provided <l> is less than e.
Lines of force whose index is greater than e are finite lines. If
e is zero, they are all finite.

The lines of force corresponding to a field of uniform force
parallel to the axis are lines parallel to the axis, the distances
from the axis being the square roots of an arithmetical series.

The theory of equipotential surfaces and lines of force in two
dimensions will be given when we come to the theory of con-
jugate functions*.

* See Â» paper ' On the Flow of ElectHdtj in Conducting Sorfaoes/ by Prof. W. B.
Snutli, Proe, Â£.8. Edin., 1869-70, p. 79.

ajfjL^/CC fi^r^-fjM, M./K4^ ^ ^f^ tL^ ^ tfi4^ s^ ^i^

JJ^ TJiuJc jls^w -//^w^^ X^^^^^^ f^.r,M.

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CHAPTER Vm.

SIMPLE CASES OF ELECTEIPICATION.

Two Parallel Planes.

124.] We shall consider, in the first place, two parallel plane
conducting surfaces of infinite extent, at a distance c firom each
other, maintained respectively at potentials A and B.

It is manifest that in this case the potential V will be a
function of the distance z from the plane A^ and will be the same
for all points of any parallel plane between A and jB, except
near the boundaries of the electrified surfaces, which by the
supposition are at an infinitely great distance from the point
considered.

Hence, Laplace's equation becomes reduced to

cPV ^

the integral of which is

r^C, + C,z;
and since when = 0, F= A, and when = c, F= jB,

For all points between the planes, the resultant intensity is
normal to the planes, and its magnitude is

C

In the substance of the conductors themselves, Ji = 0. Hence
the distribution of electricity on the first plane has a surface-
density (T, where

47r(r= jK = â€¢

C

On the other surface, where the potential is jB, the surface-

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SIMPLE GASES. PABALLEL PLANES. 187

density t/ mil be equal and opposite to (r, and

c

Let us next consider a portion of the first surface whose area
is j9, taken so that no part of 8 is near the boundary of the
surface.

The quantity of electricity on this surface is c^ = flfa-, and, by
Art. 79, the force acting on every unit of electricity is iiJ, so
that the whole force acting on the area fif, and attractrug it
towards the other plane, is

Sir Stt c*

Here the attraction is expressed in terms of the area S^ the
difference of potentials of the two surfaces (jiâ€” j5), and the dis-
tance between them o. The attraction, expressed in terms of the
charge e^^ on the area Â£â€¢, is ^ _ 2^ ,

The electric energy due to the distribution of electricity on
the area Â£i, and that on the coiTCsponding area S on the surface
B defined by projecting 8 on the surface j5 by a system of lines
of force, which in this case are normals to the plane, is

\2

= *4,

_ B {A-Bf
'4

Sir
2ir

S.^^'

= Jb,

The first of these expressions is the general expression of elec-
tric energy (Art. 84).

The second gives the energy in terms of the area, the distance,
and difference of potentials.

The third gives it in terms of the resultant force iZ, and the
volume 8c included between the areas 8 and 8f^ and shews that
the energy in unit of volume is p where Btt j9 = J2^.

The attraction between the planes is ^8^ or in other words,
there is an electrical tension (or negative pressure) equal to jp on
every unit of area.

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188 SIMPLE CASES. [125.

The fourth expression gives the energy in terms of the charge.

The fifth shews that the electrical energy is equal to the work
which would be done by the electric force if the two surfaces
were to be brought together, moving parallel to themselves, with
their electric charges constant.

To express the charge in terms of the difference of potentials,
we have 1 ^ / ^ Â»v /a Â»\

The coefficient 9 represents the charge due to a difference of
potentials equal to unity. This coefficient is called the Capacity
of the surface &y due to its position relatively to the opposite
surface.

Let us now suppose that the medium between the two surfaces
is no longer air but some other dielectric substance whose specific
inductive capacity is Ky then the charge due to a given difference
of potentials vrill be K times as great as when the dielectric ia
air, or ^^ r a Â»\

^ 4 ire ^ '

The total energy will be

The force between the surfaces will be

-^ Sir c*

2ir

Hence the force between two surfaces kept at given potentials
varies directly as K^ the specific inductive capacity of the dielec-
tric, but the force between two surfaces charged with given
quantities of electricity varies inversely as K.

Two ConcerUric Spherical Surfaces.

125.] Let two concentric spherical surfaces of radii a and 6, of
which b is the greater, be maintained at potentials A and B
respectively, then it is manifest that the potential F is a function
of r the distance from the centre. In this case, Laplace's equa-
tion becomes cPV 2 dV ^ ^
dr^ r dr ^ '

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125-] CONOENTEIO SPHEEIOAL SUEFACES. 189

The solution of this is

and the conditions that V=A when r = a, and V=^B when r = 6,
give for the space between the spherical surfaces,
^ Aa-Bb A-B .
aâ€” 6 a^ â€” 6^
dV _ A-B

If (Tj, (Tj are the sorfiace-densities on the opposed surfaces of a
solid sphere of radius a, and a spherical hollow of radius b, then
1 A-B 1 B-A

If Â«i and Bg are the whole charges of electricity on these
surfaces,

ei = 4woVi = ^=iâ€” pr = -Â«2-

jL

The capacity of the enclosed sphere is therefore r â€” â€¢

K the outer surface of the shell is also spherical and of radius c,
then, if there are no other conductors in the neighbourhood, the
charge on the outer surface is

^3 = Be,

Hence the whole charge on the inner sphere is

and that on the outer shell

^2 + ^3 = 5^(^-^) + ^^-

If we put 6 = 00, we have the case of a sphere in an infinite
space. The electric capacity of such a sphere is a, or it is

The electric tension on the inner sphere per unit of area is

The resultant of this tension over a hemisphere is ira^ = ^
normal to the base of the hemisphere, and if this is balanced by
a surface tension exerted across the circular boundary of the
hemisphere, the tension on unit of length being T, we have

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190 SIMPLE OASES. [126.

Hence ^ = -^^1 4 = ^Â»

le^ra (6â€” a)*

If a spherical soap bubble is electrified to a potential J, then,
if its radius is a, the charge will be Aa^ and the surface-density
will be 1 ^

~ 47ra *

The resultant intensity just outside the surface will be 47r(r,
and inside the bubble it is zero, so that by Art. 79 the electric
force on unit of area of the surface will be 2 irir*, acting outwarda
Hence the electrification will diminish the pressure of the air
within the bubble by 27r<r*, or

^_^

But it maybe shewn that ilT^ is the tension which the liquid
film exerts across a line of unit length, then the pressure from
within required to keep the bubble from collapsing is 2TQ/a. If
the electric force is just sufficient to keep the bubble in equi-
librium when the air within and without is at the same pressure,

Two Infinite Coaxal Cylindric Surfaces.

126.] Let the radius of the outer surface of a conducting
cylinder be a, and let the radius of an inner surface of a hollow
cylinder, having the same axis vrith the first, be 6. Let their
potentials be A and B respectively. Then, since the potential V
is in this case a function only of r, the distance from the axis,
Laplace's equation becomes

dr^ r dr '
whence F= C^ + C^ log r.

Since F= A when r^^ay and F=5 when r = 6,

h T

A\og- + J?l0flr -

y ^r ^_a

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127.]

COAXAL CTLINDBBS.

191

If a-i, ^2 are the sorface-densitiea on the inner and outer
surfifuses,

A-B . B-A

4ir<ri=-

4ir<r2 =

alog-

Mog-

If Ci and â‚¬2 are the charges on the portions of the two cylinders

between two sections transverse to the axis at a distance I from

eachother^ A-^B

Ci = 2ira2<ri = i â€” r- I = â€”e^*

The capacity of a length I of the interior cylinder is therefore

logs

If the space betwen the cylinders is occupied by a dielectric of
specific inductive capacity K instead of air, then the capacity of
a length I of the inner cylinder is

b '

The energy of the electrical distribution on the part of the
infinite cylinder which we have considered is

Kg. 5.

127.] Let there be two hollow cylindric conductors A and 5,
Fig. 5, of indefinite length, having the axis of x for their common
axis, one on the positive and the other on the negative side of
the origin, and separated by a short interval near the origin
of coordinates.

Let a cylinder C of length 22 be placed with its middle point
at a distance x on the positive side of the origin, so as to extend
into both the hollow cylinders.

Let the potential of the hollow cylinder on the positive side be

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192 SIMPLE CASES. [127.

Af that of the one on the negative side B^ and that of the internal
one (7, and let us put o for the capacity per unit of length of C
with respect to A, and p for the same quantity with respect to B.

The surface-densities of the parts of the cylinders at fixed
points near the origin and at points at given small distances
from the ends of the inner cylinder will not be affected by the
value of X provided a considerable length of the inner cylinder
enters each of the hollow cylinders. Near the ends of the hollow
cylinders, and near the ends of the inner cylinder, there will be
distributions of electricity which we are not yet able to calculate,
but the distribution near the origin will not be altered by the
motion of the inner cylinder provided neither of its ends comes
near the origin, and the distributions at the ends of the inner
cylinder will move with it, so that the only effect of the motion
will be to increase or diminish the length of those parts of the
inner cylinder where the distribution is similar to that on an
infinite cylinder.

Hence the whole energy of the system will be, so far as it
depends on x,

Q=ia(^ + Â«)(<?-^)^ + ii3(i-aj)((7-5)2 + quantities

independent of x ;
and the resultant force parallel to the axis of the cylinders since the
energy is expressed in terms of the potentials will by Art. 93 6 be

If the cylinders A and B are of equal section, a = ^, and
X ^ a{B-'A){C^i{A+B)).

It appears, therefore, that there ia a constant force acting on
the inner cylinder tending to draw it into that one of the outer
cylinders from which its potential differs most.

If (7 be numerically large and A-hB comparatively small, then
the force is approximately x = a(B-^A) C ;
so that the difference of the potentials of the two cylinders can
be measured if we can measure X, and the delicacy of the
measurement will be increased by raising (7, the potential of the
inner cylinder.

This principle in a modified form is adopted in Thomson's

The same arrangement of three cylinders may be used as a
measure of capacity by connecting B and C. If the potential of

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12 7.] COAXAL OYLINDEBS. 193

A is zero, and that of B and (7 is F, then the quantity of elec-
tricity on il wiU be ^3 = (gi3 + a(Z+a;)) F;

where q^^ is a quantity depending on the distribution of electricity
on the ends of the cylinder but not upon x^ so that by moving C
to the right till x becomes 0? + ^ the capacity of the cylinder C
becomes increased by the definite quantity af, where

1

a =

a and b being the radii of the opposed cylindric surfaces.

VOL. I.

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CHAPTEE IX.

SPHERICAL HARMONICS.

128.] The maihematical theory of spherical harmonics has
been made the subject of several special treatises. The Handbuch
der Kugelfvmctionen of Dr. E. Heine, which is the most elaborate
work on the subject, has now (1878) reached a second edition in
two volumes, and Dr. F. Neumann has published his Beitrdge
zur Theorie der Kugdfunctionen (Leipzig, Teubner, 1878). The
treatment of the subject in Thomson and Tait*s Natural Philo-
Bophy is considerably improved in the second edition (1879), and
Mr. Todhunter's Elementary Treatise on Lapla/^es FunctionSy
Lamii's Functions, and BesaeTs Functions, together with Mr.
Ferrers' Elementary Treatise on Spherical Harmonics and
sufy'ects connected with them, have rendered it unnecessary to
devote much space in a book on electricity to the purely mathe-
matical development of the subject.

I have retained however the specification of a spherical
harmonic in terms of its poles.

On Singular Points at which the Potential becomes Infinite.

129 a.] If a charge, il^, of electiicity is uniformly spread over

the surface of a sphere the coordinates of whose centre are

(a, 6, c), the potential at any point {x, y, z) outside the sphere is,

by Art. 125, a

^ F=^, (1)

where r* = (aj-a)Â« + (y-6)2 + (0-c)Â«. (2)

As the expression for V is independent of the radius of the
sphere, the form of the expression will be the same if we suppose
the radius infinitely small. The physical interpretation of the
expression would be that the charge A^ is placed on the surface
of an infinitely small sphere, which is sensibly the same as a

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SPHBEIOAL HARMONICS. 195

mathematical point. We have already (Arts. 55, 81) shewn that
there is a limit to the siirfiEkce-density of electricity, so that it is
physically impossible to place a finite charge of electricity on a
sphere of less than a certain radius.

Nevertheless, as the equation (1) represents a possible distri-
bution of potential in the space surrounding a sphere, we may
for mathematical purposes treat it as if it arose from a charge Aq
condensed at the mathematical point (a, b, c), and we may call
the point a sin gular poin t of order zero.

There are other kinds of singular points, the properties of
which we shall presently investigate, but before doing so we must
define certain expressions which we shall find useful in dealing
with directions in space, and with the points on a sphere which
correspond to them.

1296.] An axis is any definite direction in space. We may
suppose it defined by a mark made on the surface of a sphere at
the point where the radius drawn /rom the centre in the direction
of the axis meets the surface. This point is called the Pole of
the a^ . An axis has therefore one pole only, not two.

If /A is the cosine of the angle between tiie axis k and any
vector r, and if p^ ^^^ (3)

p is the resolved part of r in the direction of the axis h.

Different axes are distinguished by different suffixes, and the
cosine of the angle between two axes is denoted by X^^, where
m and n are the suffixes specifying the axis.

Differentiation with respect to an axis, A, whose direction
cosines are Z, Jf, N^ is denoted by

From these definitions it is evident that

dK ^

If we now suppose that the potential at the point (aj, y, z) due
to a singular point of any order placed at the origin is

02

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196 SPHEBICAL HARMONICS. [129C.

then if such a point be placed at the extremity of the axis A,
the potential at (x^ y, z) will be

Af[{x^Lh\ (y-Mk), {z^Nh)l

and if a point in all respects the same, except that the sign of A
is reversed, be placed at the origin, the potential due to the pair
of points will be

r=^Af[{x^Lh\ {y-Mh\ {z^Nh)]^Afix,y,zl
= -^Ah^f{x, y, z) + terms containing A*.

If we now diminish A and increase A without limit, their pro-
duct continuing finite and equal to A\ the ultimate value of the
potential of the pair of points will be

F=-il'^/(a^y.4 (8)

lif{x, y, z) satisfies Laplace's equation, then, since this equation
is linear, K, which is the difference of two functions, each of
whidi separately satisfies the equation, must itself satisfy it.

129 c.] Now the potential due to a singular point of order zero,

Vo = ^,l' (9)

satisfies Laplace's equation, therefore every function formed from
this by differentiation with respect to any number of axes in
succession must also satisfy that equation.

A point of the first oi'der may be formed by taking two points
of order zero, having equal and opposite charges â€”A^ and ^0,
and placing the first at the origin and the second at the extremity
of the axis h^ . The value of h^ is then diminished and that of
4o increased indefinitely, but so that the product Aq h^ is always
equal to Ai- The ultimate result of this process, when the two
points coincide, is a point of the first order whose moment is A^
and whose axis is hi. A point of the first order is therefore a
double point Its potential is

mt iai 18 y

(10)

By placing a point of the first order at the origin, whose
moment is â€” ili, and another at the extremity of the axis h^

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I30a.] SOLID HAEMONIO OP POSITIVE DEGREE. 197

whose moment is il^, and then diminishing h^ and increasing il^,
Â«otlÂ«* A,h,^iAâ€ž (11)

we obtain a point of the second order, whose potential is y

^ A,i'Ji!tipb.: ^ (12)

We may call a point of the second order a quadruple point
because it is constructed by making four points of order zero
approach each other. It has two axes hi and A^ and a moment
A 2. The directions of these axes and the magnitude of the
moment completely define the nature of the point.

By differentiating with respect to n axes in succession we
obtain the potential due to a point of the n*^ order. It will be
the product of three factors, a constant, a certain combination of
cosines, and r~(Â»+i). It is convenient, for reasons which will
appear as we go on, to make the numerical value of the constant
sudi that when all the axes coincide with the vector, the co-
efficient of the moment is r~<*+^). We therefore divide by n
when we differentiate with respect to h^.

In this way we obtain a definite numerical value for a par-
ticular potential, to which we restrict the name of The Solid,
Harmonic of degree â€” fa -hi ), namely

y. /i\Â» ^ d d d I ,Â»

^â€¢"^"^^ 1.2.3...ndh^'dh^'''dh,'r' ^ ^

If this quantity is multiplied by a constant it is still the
potential due to a certain point of the n^ order.

129 d.] The result of the operation (13) is of the form

lJ=i;r-<*+i), (14)

where ^ is a function of the n cosines fAi.../AÂ« of the angles
between r and the n axes, and of the in(n^ 1) cosines Aj2, &c.
of the angles between pairs of the axes.

If we consider the directions of r and the n axes as determined
by points on a spherical surface, we may regard }^ as a quantity
varying from point to point on that surface, being a function of the
I n (n + 1) distances between the n poles of the axes and the pole
of tie vector. We therefore call I^ The Surfa ^^ TTi^prmnift nf
order n.

180a,] We have next to shew that to every surface-harmonic

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198 SPHEEIOAL HAEMONICS4 [1306.

of order n there corresponds not only a solid harmonic of degree
â€” (ti + 1) but another of degree n, or that

5;, = I3;r-= TJr2"+i â€¢ (16)

satisfies Laplace's equation.

For ^= (2n+l)r2-ia?Tj:+r2Â«+i^,

'^" = (27i+l)[(2n-l)ar^ + r2]r2Â«-3]r+2(27i+l)r2â€” iflj^

^^ d^
Hence

Now, since 1^ is a homogeneous function of x, y, and z, of
negative degree Â«+ 1,

dV d^ dV , ,.rr /,,x

The first two terms therefore of the right-hand member of
equation (16) destroy each other, and, since J^ satisfies Laplace's
equation, the third term is zero, so that H^ also satisfies Laplace's
equation, and is therefore a solid harmonic of degree n.

This is a particular case of the more general theorem of
electrical inversion, which asserts that if F (x, y, z) is a function
of X, y, and z which satisfies Laplace's equation, then there exists
another function, aâ€ž^a^x a^y a^Zy.

which also satisfies Laplace's equation. See Art. 162.

1806.] The surface harmonic IJ[ contains 2n arbitrary vari-
ables, for it is defined by the positions of its n poles on the
sphere, and each of these is defined by two coordinates.

Hence the solid harmonics T^ and H^ also contain 2n arbitrary
variables. Each of these quantities, however, when multiplied
by a constant, will satisfy Laplace's equation.

To prove that AH^ is the most general rational homogeneous
function of degree n which can satisfy Laplace's equation, we

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I3I&.] SOLID HABMONIO OF POSITIVE DBGEEE. 199

observe that f, the general rational homogeneous function of
degree ti, contains J (n+ 1) (71 + 2) terms. But V^K is a homo-
geneous function of degree 71â€” 2, and therefore contains ^ 11 (71â€” 1)
terms, and the condition V^K = requires that each of these
must vanish. There are therefore 1 71(71â€” 1) equations between
the coefficients of the i(7i-f l)(n+2) terms of the function K,
leaving 271 + 1 independent constants in the most general form
of the homogeneous function of degree n which satisfies Laplace's
equation. But J?^, when multiplied by an arbitrary constant,
satisfies the required conditions, and has 2 7i + 1 arbitrary con-
stants. It is therefore of the most general form.

131a.] We are now able to form a distribution of potential
such that neither the potential itself nor its first derivatives
become infinite at any point.

The function J^ = IJjr~^"+^) satisfies the condition of vanishing
at infinity, but becomes infinite at the origin.

The function H^ = I^r* is finite and continuous at finite dis-
tances from the origin, but does not vanish at an infinite distance.

But if we make a*^r~<*+^) the potential at all points outside
a sphere whose centre is the origin, and whose radius is a, and
a-<*+*)IJir" the potential at all points vrithin the sphere, and if
on the sphere itself we suppose electricity spread with a surface
density cr such that

4w<raÂ« = (271+1)^, (18)

then all the conditions will be satisfied for the potential due to
a shell charged in this manner.

For the potential is everywhere finite and continuous, and
vanishes at an infinite distance ; its first derivatives are every-
where finite and are continuous exeept at the charged surface,
where they satisfy *the equation

av dv

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