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function which we, following Thomson and Tait, denote by 0.

The function which Heine {HaTidbuch der KiLgd/unctumeny
§ 47) denotes by j^'*^ and calls eine zugeordnete Function erster
Art, or, as Todhunter translates it, an ' Associated Function of
the First Blind,' is related to 0|^^ by the equation

0^'> = (-l)^^<">. (76)

The series of descending powers of /i, beginning with /i*~^, is

expressed by Heine by the symbol 5p^"\ and by Todhunter by the

symbol «r (a, n).

This series may also be expressed in two other forms,

_ 2-{n-a)\n\ d- . .

- (2nj\ d^^""' ^ ^
The last of these, in which the series is obtained by differentiating
the zonal harmonic with respect to ii, seems to have suggested the
symbol T^^^ adopted by Ferrers, who defines it thus

2<-)-^*lp-_ii!^)i_0<'>. (77)

When the same quantity is expressed as a homogeneous
function of /ut and y, and divided by the coefficient of m*""' ^^'j ^^
is what we have already denoted by ^^.



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214 SPHERIOAL HABKONIOS. [14I.

140c.] The harmonics of the symmetrical system have been
classified by Thomson and Tait with reference to the form of the
spherical curves at which they become zero.

The value of the zonal harmonic at any point of the sphere is
a function of the cosine of the polar distance, which if equated
to zero gives an equation of the n^^ degree, all whose roots lie
between — 1 and + 1, and therefore correspond to n parallels of
latitude on the sphere.

The zones included betwete these parallels are alternately
positive and negative, the circle surrounding the pole being
always positive.

The zonal harmonic is therefore suitable for expressing a
function which becomes zero at certain parallels of latitude on
the sphere^ or at certain conical surfaces in spaca

The other harmonics of the symmetrical system odsur in pairs,
one involving the cosine and the other the sine of o-<^. They
therefore become zero at <r meridian circles on the sphere and
also at 71— 0- parallels of latitude, so that the spherical surface is
divided into 20- (71— o-— 1) quadrilaterals or tesserae, together with
4 o- triangles at the poles. They are therefore useful in investiga-
tions relating to quadrilaterals or tesserae on the sphere bounded
by meridian circles and parallels of latitude.

They are all called Tesseral harmonics except the last pair,
which becomes zero at n meridian circles only, which divide the
spherical surface into 2 n sectors. This pair are therefore called
Sectorial harmonics.

141.] We have next to find the surface integral of the square of
any tesseral harmonic taken over the sphere. This we may do by
the method of Art. 134. We convert the surface harmonic F^'^
into a solid haimonic of positive degree by multiplying it by r".
We differentiate this solid harmonic with respect to the n axes of
the harmonic itself, and then make a: = j/ = 2; = 0, aiid we

A 2

multiply the result by —7-7- -^ •

^•^ -^711(271+1)

These operations are indicated in our notation by

Writing the solid harmonic in the form of a homogeneous
function of z and f and 1;, viz..



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142 &.] SITBPAOB INTBGBALS, 215

we find that on performing the differentiations with respect
to 0, all the terms of the series except the first disappear, and
the factor (Ti—a) I is introduced.

Continuing the differentiations with respect to f and i; we
get rid also of these variables and introduce the factor — 2i 0*1, so
that the final result is

JJ^ n^ 271+1 2^^n\n\ ^ '

We shall denote the second member of this equation by the
abbreviated symbol [?i, o-].

This expression is correct for all values of <r from 1 to ti inclu-
sive, but there is no harmonic in sin o-^ corresponding to o- = 0.
In the same way we can shew that

J J K^V "^ - 271+ 1 2^''7ll7ll ^^^f

for aU values of a from 1 to ti inclusive.

When o- = 0, the harmonic becomes the zonal harmonic, and

a result which may be obtained directly from equation (50) by
putting I^ = i^ and remembering that the value of the zonal
harmonic at its pole is unity.

142 a.] We can now apply the method of Art. 136 to determine
the coefficient of any given tesseral surface harmonic in the
expansion of any arbitrary function of the position of a point on

a sphere. For let F be the arbitrary function, and let -4^ be the
coefficient of Y^J^ in the expansion of this function in surface
harmonics of the symmetrical system, then

ffFy:'ds = A^'Jf(Yt')'d8 = 4'>[n, «r] . (83)

where [71, a] is the abbreviation for the value of the surface in-
tegral given in equation (80).

1426.] Let ^ be any function which satisfies Laplace's equa-
tion, and which has no singular values within a distance a of a
point 0, which we may take as the origin of coordinates. It
is always possible to expand such a function in a^ series of solid
harmonics of positive degree, having their origin at 0.



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216 SPHERICAL HABM0NI08. [l43-

One way of doing this is to describe a sphere about as centre
with a radius less than a, and to expand the value of the potential
at the surface of the sphere in a series of surface harmonics.
Multiplying each of these harmonics by r/a raised to a power
equal to the order of the surface harmonic, we obtain the solid
harmonics of which the given function is the sum.

But a more convenient method, and one which does not involve
integration, is by differentiation with respect to the axes of the
harmonics of the symmetrical system.

For instance^ let us suppose that in the expansion of % there is

(<r) (<r)

a term of the form Ac Yc r\

n n

K we perform on * and on its expansion the operation
and put X, y, z equal to zero after differentiating, all the terms

(<r)

of the expansion vanish except that containing Ac,

Expressing the operator on 4^ in terms of differentiations with
respect to the real axes, we obtain the equation

dz'-' Ida^ 1 . 2 diC-* dy^ ^ J

= i?(!i±4fc^% (84)

from which we can determine the coefficient of any harmonic
of the series in terms of the differential coefficients of ^ with
respect to x^yyZ at the origin.

143.] It appears from equation (50) that it is always possible
to express a harmonic as the sum of a system of zonal harmonics
of the same order, having their poles distributed over the surface
of the sphere. The simplification of this system, however, does
not appear easy. I have, however, for the sake of exhibiting to
the eye some of the features of spherical harmonics, calculated
the zonal harmonics of the third and fourth orders, and drawn, by
the method already described for the addition of functions, the
equipotential lines on the sphere for harmonics which are the
sums of two zonal harmonics. See Figures VI to IX at the end
of this volume.

Fig. VI represents the difference of two zonal harmonics of the
third order whose axes are inclined at 120** in the plane of the



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I44<*-] DIAGRAMS OF SPHERICAL HARMONICS. 217

paper, and this difference is the harmonic of the second type in
which <r = 1, the axis being perpendicular to the paper.

In Fig. VII the harmonic is also of the third order, but the
axes of the zonal harmonics of which it is the sum are inclined at
90°, and the result is not of any type of the symmetrical system.
One of the nodal lines is a great circle, but the other two which
are intersected by it are not circles.

Fig. Yin represents the difference of two zonal harmonics of
the fourth order whose axes are at right angles. The result is a
tesseral harmonic for which n = 4, o- = 2.

Fig. IX represents the sum of the same zonal harmonics. The
result gives some notion of one type of the more general har-
monic of the fourth order. In this type the nodal line on the
sphere consists of six ovals not intersecting each other. Within
these ovals the harmonic is positive, and in the sextuply con-
nected part of the spherical surface which lies outside the ovals,
the harmonic is negative.

All these figures are orthogonal projections of the spherical
surface.

I have also drawn in Fig. V a plane section through the axis
of a sphere, to shew the equipotential surfaces and lines of force
due to a spherical surface electrified according to the values of a
spherical harmonic of the first order.

Within the sphere the equipotential surfaces are equidistant
planes, and the lines of force are straight lines parallel to the
axis, their distances from'the axis being as the square roots of the
natural numbers. The lines outside the sphere may be taken as
a representation of those which would be due to the earth's mag-
netism if it were distributed according to the most simple tjrpe.

144 a.] We are now able to determine the distribution of
electricity on a spherical conductor under the action of electric
forces whose potential is given.

By the methods already given we expand ^, the potential due
to the given forces, in a series of solid harmonics of positive
degree having their origin at the centre of the sphere.

Let il„r*I^ be one of these, then since within the conducting
sphere the potential is uniform, there must be a term — il^r*]^
arising from the distribution of electricity on the surface of the
sphere, and therefore in the expansion of 4 inr there must be a
term ^ita^ = {2n+ l)a*~i A^Y^.



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218 SPHBEIOAL HABMONICS. [^44 &•

In this way we can determine the coefficients of the harmonics
of all orders except zero in the expression for the surface density.
The coefficient corresponding to order zero depends on the charge,
e, of the sphere, and is given by 4ir<ro = a~*c.

The potential of the sphere is

** a

144 &.] Let us next suppose that the sphere is placed in the
neighbourhood of conductors connected with the earth, and that
Green's Function, (?, tss been determined in terms of a;, y, z and
^') j/} ^i the coordinates of any two points in the region in which
the sphere is placed.

K the surface density on the sphere is expressed in a series
of spherical harmonics, then the electrical phenomena outside the
sphere, arising from this charge on the sphere, are identical with
those arising from an imaginary series of singular points all
at the centre of the sphere, the first of which is a single point
having a charge equal to that of the sphere and the others are
multiple points of difierent orders corresponding to the harmonics
which express the surface density.

Let Green's function be denoted by Gpj/, where p indicates the
point whose coordinates ai-e x, y, z, and p' the point whose co-
ordinates are x\ y^, z\

If a charge Aq is placed at the point p\ then, considering
aj', y', / as constants, G^^f becomes a function of x,y,z\ and the
potential arising from the electricity induced on surrounding
bodies by il^ is * = Af^G^^/. (1)

If, instead of placing the charge Aq at the point p\ it were
distributed uniformly over a sphere of radius a having its centre
at p\ the value of ^ at points outside the sphere would be the
same.

If the charge on the sphere is not uniformly distributed, let
its surface density be expressed, as it always can, in a series of
spherical harmonics, thus

^Tta^a^z AQ-\-SAiYi-^&c. + {2n+l)AJ'^+.... (2)

The potential arising from any term of this distribution, say

47raV^ = (27i+l)^X i^)

will be — ;^ A^y^ for points inside the sphere, and -^^ A^T^ for
points outside the sphere.



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144 &•] gbbbn's PUNOTioisr. 219

Now the latter expression, by equations (13), (14), Arts. 129 c
and 1 29 d is equal to . v^ . a* d* 1

or the potential outside the sphere, due to the charge on the
surface of the sphere, is equivalent to that due to a certain
multiple point whose axes are Ai . . . A^ and whose moment is

Hence the distribution of electricity on the surrounding con-
ductors and the potential due to this distribution is the same as
that which would be due to such a multiple point.

The potential, therefore, at the point jt>, or (aj, y, «), due to the
induced electrification of surrounding bodies, is

where the accent over the c2's indicates that the differentiations
are to be performed with respect to x\ y", z\ These coordinates are
afterwards to be made equal to those of the centre of the sphere.
It is convenient to suppose Y^ broken up into its 27i+ 1 con-
stituents of the symmetrical system. Let ^^ Y^"^ be one of
these, then d'* . , i^s.

It is unnecessary here to supply the affix a or c, which indicates
whether sin o-^ or cos <r^ occurs in the harmonic.

We may now write the complete expression for *, the potential
arising from induced electrification,

♦ = ^G'+22[(-l)M^"^"i)^')(?] . (6)

But within the sphere the potential is constant, or

♦ + 1^0+22 [^^|;;.>1^.>]= constant. (7)

Now perform on this expression the operation D^^ , where the
difierentiations are to be with respect to a;, y^ 0, and the values
of n^ and o-} are independent of those of n and a. All the terms
of (7) will disappear except that in Y^^^ and we find

g (7h + <ri)!(^~<ri)l 1 .<o
2^1 Til I a«»+i \

= ^2)<^>(?+22[(-l)«< Jj2)i^^>^^^^^ (8)

We thus obtain a set of equations, the first member of each of



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220 SPHERICAL HARMONICS. [l45 «.

which contains one of the coefficients which we wish to deter-
mine. The first term of the second member contains A^, the
charge of the sphere, and we may regard this as the principal
term.

Neglecting, for the present, the other terms, we obtain as a
first approximation

**» 2(7ii + cri)!('ni-(ri)I ^ \ ^ ^

If the shortest distance from the centre of the sphere to the
nearest of the surrounding conductors is denoted by 6,

If, therefore, b is large compared with a, the radius of the
sphere, the coefficients of the other spherical harmonics are very
small compared with -4^,. The ratio of a term after the first on
the right-hand side of equation (8) to the first term will there-

fore be of an order of magnitude similar to (r)

We may therefore neglect them in a first approximation, and
in a second approximation we may insert in these terms the
values of the coefficients obtained by the first approximation,
and so on till we arrive at the degree of approximation required.

Distribution of electricity on a nearly spherical conductor.

145 a.] Let the equation of the surface of the conductor be

r = a(I+i7, (1)

where ^ is a function of the direction of r, that is to say of
and ^, and is a quantity the square of which may be neglected
in this investigation.

Let ^ be expanded in the form of a series of surface harmonics

^=/o+/iir+/2i$+&c-+/x (2)

Of these terms, the first depends on the excess of the mean
radius above a. If therefore we assume that a is the mean
radius, that is to say approximately the radius of a sphere whose
volume is equal to that of the given conductor, the coefficient /^
will disappear.

The second term, that in /j, depends on the distance of the
centre of mass of the conductor, supposed of uniform density,



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I45«-] NBAELY 8PHBEICAL CONDUOTOBS. 221

from the origin. If therefore we take that centre for origin, the
coefficient /i will also disappear.

We shall begin by supposing that the conductor has a charge
Aq , and that no external electrical force acts on it. The potential
outside the conductor must therefore be of the form

r=A, I +^,^'i +&c.+^X^+.... (3)

where the surface harmonics are not assumed to be of the same
types as in the expansion of F.

At the surface of the conductor the potential is that of the
conductor, namely, the constant quantity a.

Hence, expanding the powers of r in terms of a and jP, and
n^lecting the square and higher powers of jP, we have

+A,±.j:il-(n+l)]l)+.... (i)

Since the coefficients il^, &c. are evidently small compared
with Aq^ we may begin by neglecting products of these co-
efficients into F.

If we then write for F in its first term its expansion in
spherical harmonics, and equate to zero the terms involving
harmonics of the same order, we find

« = A^' (6)

A,Ti^=A,af,X=0, (6)



AJ:' = Aoa'f,7,. (7)

It follows from these equations that the Vb must be of the

same type as the Fs, and therefore identical with them, and

that -4.1 = and A^ = A^a^ f^.

To determine the density at any point of the surface, we have

the equation ^ dV dV • x i /«x

^ 4ircr = —-7- = — -r~^8 ^> approximately ; (8)

where v is the normal and e is the angle which the normal makes
with the radius. Since in this investigation we suppose F and
its first differential coefficients with respect to and (^ to be
small, we may put cos e = 1, so that

4^<r=-^ = ^„l+&c. + («+l)^.^^,+ .... (9)



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222 SPHBBIOAL HABH0NIC8. [l45&*

Expanding the powers of r in terms of a and F, and neglecting
products of F into -4.^, we find

4,r(r=iloi(l-2^ + 8w5. + («+l)^.^,^. (10)

Expanding jPin spherical harmonics and giving A^ its value
as already found, we obtain

4^<r=^i[l+AIJ+2/3^+&0. + (n-l)/.^. (11)

Hence, if the surface differs from that of a sphere by a thin
stratum whose depth varies according to the values of a spherical
harmonic of order ti, the ratio of the difference of the surface
densities at any two points to their sum will be 7i~l times
the ratio of the difference of the radii at the same two points to
their sum.

145 &.] If the nearly spherical conductor (1) is acted on by
external electric forces, let the potential, Uy arising from these
forces be expanded in a series of spherical harmonics of positive
degree, having their origin at the centre of volume of the
conductor

?7' = 5o+^i^ir'+^8^5J'+&c. + 5^r*i;;'+..., (12)

where the accent over Y indicates that this harmonic is not
necessarily of the same type as the harmonic of the same order
in the expansion of F.

If the conductor had been accurately spherical, the potential
arising from its surface charge at a point outside the conductor
would have been

F= A,\-B,^X'-&c.-B,''^T:-.... (13)

Let the actual potential arising from the surface charge be
F+ TT, where

Tr = C,iF/' + &c. + (7«^F."+...; (14)

the harmonics with a double accent being different from those
occurring either in F or in U^ and the coefficients C being small
because F is small.

The condition to be fulfilled is that, when r = a (1 +^,

Cr+ F+ F = constant = Aq - ¥B^,

the potential of the conductor.



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H5^-] NEARLY SPHEBIOAL VESSELS. 223

Expanding the powers of r in terms of a and F, and retaining
the first power of ^ when it is multiplied by A or B, but neglect-
ing it when it is multiplied by the small quantities (7, we find

+ C,ir + &c. + ^«^^"+...= 0. (15)

To determine the coefficients C,we must perform the multipli-
cation indicated in the first line, and express the result in
a series of spherical harmonics. This series, with the signs
reversed, will be the series for W at the surface of the con-
ductor.

The product of two surface spherical harmonics of orders n
and m, is a rational function of degree n + m in. a/r, y/r, and z/r,
and can therefore be expanded in a series of spherical harmonics
of orders not exceeding m + ti. If, therefore, F can be expanded
in spherical harmonics of orders not exceeding m, and if the
potential due to external forces can be expanded in spherical
harmonics of orders not exceeding n, the potential arising from
the surface charge will involve spherical harmonics of orders
not exceeding m + n.

This surface density can then be found from the potential by
the approximate equation

47r(r+^([7"+r+Tr) = 0. (16)

145 c.] A nearly spherical conductor enclosed in a nearly
spherical and nearly concentric conducting vessel.

Let the equation of the surface of the conductor be

r = a(l+^, (17)

where ^=/i Jr+&c.+/^'> y^^^^^. (18)

Let the equation of the inner surface of the vessel be

r = 6(1 + 0), (19)

where » = fl^i 3?+ &c. +g^;> y^<'>, (20)

the/*8 and gr's being small compared with unity, and F; being
the surface harmonic of order n and type <r.

Let the potential of the conductor be a, and that of the
vessel )9. Let the potential at any point between the conductor
and the vessel be expanded in spherical harmonics, thus



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224 SPHEBIOAL EABHONIOS. [146.

then we have to determine the constants of the forms h and k so
that when r=a(l +i^, 4' = a, and when r = 6(1 + G), 4^ = /3.

It is manifest, from our former investigation, that all the JCs
and A;'s except h^ and k^ will be small quantities, the products of
which into F may be neglected. We may, therefore, write

« = *o+*o^(l-fO+&c.+ (AWa- + A;W_l^)y<')+..., (22)

We have therefore , , 1 ,„ .

« = ^+^o^' (24)

fi = K + K\, (25)

*o^yi'> = A::^a"+Ar^. (26)

whence we find for k^, the charge of the inner conductor,

*o = («-i8)5??^„. (28)

and for the coefficients of the harmonics of order n



h^:^=klIlL—lil^,



^6an+l_^2n+l



(29)



where we must remember that the coefficients f^\ ^^\ h^^\ k^^^ are
those belonging to the same type as well as order.

The surface density on the inner conductor is given by the
equation

_ /;^{(^ + 2)^'"^^ +(^- l)fe'^^^ } -9n\^n+ 1) a^^'b- ^ ,3^.
where il„— j!>aii+i_^2n+i ' ^ ^

146.] As an example of the application of zonal harmonics,
let us investigate the equilibrium of electricity on two spherical
conductors.



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146.] TWO SPHBfilCAL OONDUCTOBS, 226

Let a and h be the radii of the spheres, and c the distance
between their centres. We shall also, for the sake of brevity,
write a = cx^ and & = c^^ so that x and y are numerical quantities
less than unity.

Let the line joining the centres of the spheres be taken as
the axis of the zonal harmonics, and let the pole of the zonal
harmonics belonging to either sphere be the point of that sphere
nearest to the other.

Let r be the distance of any point from the centre of the first
sphere, and 8 the distance of the same point from that of the
second sphere.

Let the surface density, o-i, of the first sphere be given by the
equation

4ir(ria*=:il + iliiJ+3ilj-^4-&c. + (2m+l)il^ii, (1)

so that A is the total charge of the sphere, and A^, &c. are the
coefficients of the zonal harmonics P^, &o.

The potential due to this distribution of charge may be repre-
sented by

U'=\[A^A,P:-^A,^^,^^^A^P,g\ (2)

for points inside the sphere^ and by

D'=i[il + ^i,^+A^^V&c. + .l.P.^] (3)

for points outside.

Similarly, if the surface density on the second sphere 'is given
by the equation

^Tta^lfl = 5+ 5i^+&c. + (27i+ 1)5^^, (4)

the potential inside and outside this sphere due to this charge
may be represented by equations of the form

F'=i[5 + £i^| + &c. + B.P.i;j, (5)

F = l[£+5,^^ + &c.+B.P.^;]. (6)

where the several harmonics are related to the second sphere.

The charges of the spheres are A and B respectively.

The potential at every point within the first sphere is constant
and equal to a, the potential of that sphere, so that within the
first sphere tT' + F = a. (7)

VOL. I. Q



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226



SPHEBIGAL HABMONICS.



[146.



Similarly, if the potential of the second sphere is /3, for points
vrithin that sphere, jj^ V'^, ff. (8)

For points outside both spheres the potential is % where

U+r=*. (9)

On the axis, between the centres of the spheres,

r-^a^c. (10)

Hence, differentiating with respect to r, and after differentiation
making r = 0, and remembering that at the pole each of the
zonal harmonics is unity, we find



(H)



A


1


dV
da ~


0.


A


- +


dii'~


0.




A


• ■

ml


• *


d'V



*«^.m+l



where, after differentiation, a is to be made equal to c.

If we perform the differentiations, and write a/c = x and
h/c = y, these equations become

= ilj + J5a;3 + 35i»3y + 6^2^? V + &c. + i (71 + 1) (ti + 2) 5^aj V»



= il^ + 5a;-»-i + (m+l)5ia;"-*-iy + 4(^^+0(^ + 2)52aj"'+V

ml 71 1 ' '

By the correspondiog operations for the second sphere we find,

= Bi + Ay^+3AiXy'^+6A^as^^ + gm. + i(vi+l){m+2)A^x''y\

= B, + ily"+H (to + 1) Aa^"** + K« + 0(« + 2)-4ja!*y"+^ + &c.

(m+^ ,

ml 71 1 ^ /



(12)



(13)



To determine the potentiab, a and /3, of the two spheres we
have the equations (7) and (8), which we may now write

ca=zAl + B + B,y + B^y^ + kc. + B,y-, (14)

cp=zB - {-A + A^x + A^ix? + &c,'i-A^x'^. (15)



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146,] TWO SPHBEIOAL CONDUOTOES. 227



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