A Treatise on Electricity and Magnetism/Volume 1/Part 1/Chapter 9

by James Clerk Maxwell

CHAPTER IX.

SPHERICAL HARMONICS.

On Singular Points at which the Potential becomes Infinite.

128.] We have already shewn that the potential due to a quantity of electricity $e$, condensed at a point whose coordinates are (a, b, c) is

$V={\frac {e}{r}}$

(1)

where r is the distance from the point (a, b, c) to the point (x, y, z), and V is the potential at the point (x, y, z).

At the point (a, b, c) the potential and all its derivatives become infinite, but at every other point they are finite and continuous, and the second derivatives of $V$ satisfy Laplace's equation.

Hence, the value of $V$, as given by equation (1), may be the actual value of the potential in the space outside a closed surface surrounding the point (a, b, c), but we cannot, except for purely mathematical purposes, suppose this form of the function to hold up to and at the point (a, b, c) itself. For the resultant force close to the point would be infinite, a condition which would necessitate a discharge through the dielectric surrounding the point, and besides this it would require an infinite expenditure of work to charge a point with a finite quantity of electricity.

We shall call a point of this kind an infinite point of degree zero. The potential and all its derivatives at such a point are infinite, but the product of the potential and the distance from the point is ultimately a finite quantity $e$ when the distance is diminished without limit. This quantity $e$ is called the charge of the infinite point.

This may be shewn thus. If $V'$ be the potential due to other electrified bodies, then near the point $V'$ is everywhere finite, and the whole potential is

$V=V'+{\frac {e}{r}},$

whence

$Vr+V'r+e.\,$

When $r$ is indefinitely diminished $V'$ remains finite, so that ultimately

$Vr=e.\,$

129.] There are other kinds of singular points, the properties of which we shall now investigate, but, before doing so, we must define some expressions which we shall find useful in emancipating our ideas from the thraldom of systems of coordinates.

An axis is any definite direction in space. We may suppose it defined in Cartesian coordinates by its three direction-cosines l, m, n, or, better still, we may suppose a mark made on the surface of a sphere where the radius drawn from the centre in the direction of the axis meets the surface. We may call this point the Pole of the axis. An axis has therefore one pole only, not two.

If through any point x, y, z a plane be drawn perpendicular to the axis, the perpendicular from the origin on the plane is

$p=lx+my+nz.\,$

(2)

The operation

${\frac {d}{dh}}=l{\frac {d}{dx}}+m{\frac {d}{dy}}+n{\frac {d}{dz}},$

(3)

is called Differentiation with respect to an axis $h$ whose direction-cosines are l, m, n.

Different axes are distinguished by different suffixes.

The cosine of the angle between the vector $r$ and any axis $h_{i}$ is denoted by $\lambda _{i}$ and the vector resolved in the direction of the axis $p_{i}$, where

$\lambda _{i}=l_{i}x+m_{i}y+n_{i}z=p_{i}.$

(4)

The cosine of the angle between two axes $h_{i}$ and $h_{j}$ is denoted by $\mu _{ij}$ where

$\mu _{ij}=l_{i}l_{j}+m_{i}m_{j}+n_{i}n_{j}.$

(5)

From these definitions it is evident that

${\frac {dr}{dh_{i}}}={\frac {p_{i}}{r}}=\lambda _{i},$

(6)

${\frac {dp_{j}}{dh_{i}}}=\mu _{ij}={\frac {dp_{i}}{dh_{j}}},$

(7)

${\frac {d\lambda _{i}}{dh_{j}}}={\frac {\mu _{ij}-\lambda _{i}\lambda _{j}}{r}}$

(8)

Now let us suppose that the potential at the point (x, y, z) due to a singular point of any degree placed at the origin is

$Mf(x,y,z).\,$

If such a point be placed at the extremity of the axis $h$, the potential at (x, y, z) will be

$Mf\left((x-lh),\ (y-mh),\ (z-nh)\right)\,;$

and if a point in all respects equal and of opposite sign be placed at the origin, the potential due to the pair of points will be

${\begin{array}{ll}V&=Mf\left\{(x-lh),\ (y-mh),\ (z-nh)\right\}-Mf(x,y,z),\\\\&=-Mh{\frac {d}{dh}}F(x,y,z)+\mathrm {terms\ containing} \ h^{2}\end{array}}$

If we now diminish $h$ and increase $M$ without limit, their product $Mh$ remaining constant and equal to $M'$, the ultimate value of the potential of the pair of points will be

$V'=-M'{\frac {d}{dh}}f(x,y,z).$

(9)

If f(x, y, z) satisfies Laplace’s equation, then $V'$, which is the difference of two functions, each of which separately satisfies the equation, must itself satisfy it.

If we begin with an infinite point of degree zero, for which

$V_{0}=M_{0}{\frac {1}{r}}$

(10)

we shall get for a point of the first degree

${\begin{array}{ll}V_{1}&=-M_{1}{\frac {d}{dh_{1}}}{\frac {1}{r}},\\\\&=M_{1}{\frac {p_{1}}{r^{3}}}=M_{1}{\frac {\lambda _{1}}{r^{2}}}\end{array}}$

A point of the first degree may be supposed to consist of two points of degree zero, having equal and opposite charges $M_{0}$ and $-M_{0}$, and placed at the extremities of the axis $h$. The length of the axis is then supposed to diminish and the magnitude of the charges to increase, so that their product $M_{0}h$ is always equal to $M_{1}$. The ultimate result of this process when the two points coincide is a point of the first degree, whose moment is $M_{1}$ and whose axis is $h_{1}$. A point of the first degree may therefore be called a Double point.

By placing two equal and opposite points of the first degree at the extremities of the second axis $h_{2}$, and making $M_{1}h_{2}=M_{2}$, we get by the same process a point of the second degree whose potential is

${\begin{array}{ll}V_{2}&=-h_{2}{\frac {d}{dh_{2}}}V_{1},\\\\&=M_{2}{\frac {d^{2}}{dh_{1}dh_{2}}}{\frac {1}{r}},\\\\&=M_{2}{\frac {3\lambda _{1}\lambda _{2}-\mu _{12}}{r^{3}}}.\end{array}}$

(12)

We may call a point of the second degree a Quadruple point, because it is constructed by making four points approach each other. It has two axes, $h_{1}$ and $h_{2}$, and a moment $M_{2}$. The directions of these two axes and the magnitude of the moment completely define the nature of the point.

130.] Let us now consider an infinite point of degree $i$ having $i$ axes, each of which is defined by a mark on a sphere or by two angular coordinates, and having also its moment $M_{i}$, so that it is defined by $2i+1$ independent quantities. Its potential is obtained by differentiating $V_{0}$ with respect to the $i$ axes in succession, so that it may be written

$V_{i}=(-1)^{i}M_{i}{\frac {d^{i}}{dh_{1}\dots dh_{i}}}\cdot {\frac {1}{r}}.$

(13)

The result of the operation is of the form

$V_{i}=|{\underline {i}}M_{i}{\frac {Y_{i}}{r^{i+1}}},$

(14)

where $Y_{1}$ it which is called the Surface Harmonic, is a function of the $i$ cosines, $\lambda _{i}\dots \lambda _{i}$ of the angles between $r$ and the $i$ axes, and of the ${\tfrac {1}{2}}i(i-1)$ cosines, $\mu _{12}$, &c. of the angles between the different axes themselves. In what follows we shall suppose the moment $M_{i}$ unity.

Every term of $Y_{i}$ consists of products of these cosines of the form

$\mu _{12}\cdot \mu _{34}\dots \mu _{2s-1\cdot 2s}\lambda _{2s+1}\dots \lambda _{i},$

in which there are $s$ cosines of angles between two axes, and $i-2s$ cosines of angles between the axes and the radius vector. As each axis is introduced by one of the $i$ processes of differentiation, the symbol of that axis must occur once and only once among the suffixes of these cosines.

Hence in every such product of cosines all the indices occur once, and none is repeated.

The number of different products of $s$ cosines with double suffixes, and $i-2s$ cosines with single suffixes, is

$N={\frac {|{\underline {i}}}{2^{s}\ {\underline {s}}\ |{\underline {i-2s}}}}$

(15)

For if we take any one of the $N$ different terms we can form from it $2^{s}$ arrangements by altering the order of the suffixes of the cosines with double suffixes. From any one of these, again, we can form $|{\underline {s}}$ arrangements by altering the order of these cosines, and from any one of these we can form $|{\underline {i-2s}}$ arrangements by altering the order of the cosines with single suffixes. Hence, without altering the value of the term we may write it in $2^{s}\ {\underline {s}}\ |{\underline {i-2s}}$ different ways, and if we do so to all the terms, we shall obtain the whole permutations of $i$ symbols, the number of which is $|{\underline {i}}$. Let the sum of all terms of this kind be written in the abbreviated form

$\sum \left(\lambda ^{i-2s}\mu ^{s}\right).$

If we wish to express that a particular symbol $j$ occurs among the $\lambda$’s only, or among the $\mu$’s only, we write it as a suffix to the $\lambda$ or the $\mu$. Thus the equation

$\sum \left(\lambda ^{i-2s}\mu ^{s}\right)=\sum \left(\lambda _{j}^{i-2s}\mu ^{s}\right)+\sum \left(\lambda ^{i-2s}\mu _{j}^{s}\right)$

(16)

expresses that the whole system of terms may be divided into two portions, in one of which the symbol $j$ occurs among the direction-cosines of the radius vector, and in the other among the cosines of the angles between the axes.

Let us now assume that up to a certain value of $i$

${\begin{array}{ll}Y_{i}=A_{i.0}\sum \left(\lambda ^{i}\right)&+A_{i.1}\sum \left(\lambda ^{i-2}\mu ^{1}\right)+\mathrm {etc} .\\\\&+A_{i.s}\sum \left(\lambda ^{i-2s}\mu ^{s}\right)+\mathrm {etc} .\end{array}}$

(17)

This is evidently true when $i=1$ and when $i=2$. We shall shew that if it is true for $i$ it is true for $i+1$. We may write the series

$Y_{i}=S\left\{A_{i.s}\sum \left(\lambda ^{i-2s}\mu ^{s}\right)\right\},$

(18)

where $S$ indicates a summation in which all values of $s$ not greater than ${\tfrac {1}{2}}i$ are to be taken.

Multiplying by $|{\underline {i}}\ r^{-(i+1)}$, and remembering that $p_{i}=r\lambda _{i}$, we obtain by (14), for the value of the solid harmonic of negative degree, and moment unity,

$V_{i}=|{\underline {i}}\ S\left\{A_{i.s}r^{2s-2i-1}\sum \left(p^{i-2s}\mu ^{s}\right)\right\}$

(19)

Differentiating $V_{i}$ with respect to a new axis whose symbol is $j$, we should obtain $V_{i+1}$ with its sign reversed,

$-V_{i+1}=|{\underline {i}}\ S\left\{A_{i,s}(2s-2i-1)r^{2s-2i-3}\sum \left(p_{j}^{i-2s+1}\mu ^{s}\right)+A_{i,s}r^{2s-2i-1}\sum \left(p^{i-2s-1}\mu _{j}^{s+1}\right)\right\}.$

If we wish to obtain the terms containing $s$ cosines with double suffixes we must diminish $s$ by unity in the second term, and we find

$-V_{i+1}=|{\underline {i}}\ S\left\{r^{2s-2i-3}\left[A_{i,s}(2s-2i-1)\sum \left(p_{j}^{i-2s+1}\mu ^{s}\right)+A_{i,s-1}\sum \left(p^{i-2s+1}\mu _{j}^{s}\right)\right]\right\}.$

(21)

If we now make

$A_{i,s}(2s-2i-1)=A_{i,s-1}=-(i+1)A_{i+1.s}$

(22)

then

$V_{i+1}=|{\underline {i+1}}S\left\{A_{i+1.s}r^{2s-2(i+1)-1}\sum \left(p^{i+1-2s}\mu ^{s}\right)\right\},$

(23)

and this value of $V_{i+1}$ is the same as that obtained by changing $i$ into $i+1$ in the assumed expression, equation (19), for $V_{i}$. Hence the assumed form of $V_{i}$, in equation (19), if true for any value of $i$, is true for the next higher value.

To find the value of $A_{i.s}$, put $s=0$ in equation (22), and we find

$A_{i+1.0}={\frac {2i+1}{i+1}}A_{i.0};$

(24)

and therefore, since $A_{i.0}$ is unity,

$A_{i.0}={\frac {|{\underline {2i}}}{2^{i}\ |{\underline {i}}\ |{\underline {i}}}};$

(25)

and from this we obtain, by equation (22), for the general value of the coefficient

$A_{i,s}=(-1)^{s}{\frac {|{\underline {2i-2s}}}{2^{i-s}\ |{\underline {i}}\ |{\underline {i-s}}}};$

(26)

and finally, the value of the trigonometrical expression for $Y_{i}$ is

$Y_{i}=S\left\{(-1)^{s}{\frac {|{\underline {2i-2s}}}{2^{i-s}\ |{\underline {i}}\ |{\underline {i-s}}}}\sum \left(\lambda ^{i-2s}\mu ^{s}\right)\right\}$

(27)

This is the most general expression for the spherical surface-harmonic of degree $i$. If $i$ points on a sphere are given, then, if any other point $P$ is taken on the sphere, the value of $Y_{i}$ for the point $P$ is a function of the $i$ distances of $P$ from the $i$ points, and of the ${\frac {1}{2}}i(i-1)$ distances of the $i$ points from each other. These $i$ points may be called the Poles of the spherical harmonic. Each pole may be defined by two angular coordinates, so that the spherical harmonic of degree $i$ has $2i$ independent constants, exclusive of its moment, $M_{i}$.

131.] The theory of spherical harmonics was first given by Laplace in the third book of his Mécanique Celeste. The harmonics themselves are therefore often called Laplace’s Coefficients.

They have generally been expressed in terms of the ordinary spherical coordinates $\theta$ and $\phi$, and contain $2i+1$ arbitrary constants. Gauss appears^[1] to have had the idea of the harmonic being determined by the position of its poles, but I have not met with any development of this idea.

In numerical investigations I have often been perplexed on account of the apparent want of definiteness of the idea of a Laplace’s Coefficient or spherical harmonic. By conceiving it as derived by the successive differentiation of ${\tfrac {1}{r}}$ with respect to $i$ axes, and as expressed in terms of the positions of its $i$ poles on a sphere, I have made the conception of the general spherical harmonic of any integral degree perfectly definite to myself, and I hope also to those who may have felt the vagueness of some other forms of the expression.

When the poles are given, the value of the harmonic for a given point on the sphere is a perfectly definite numerical quantity. When the form of the function, however, is given, it is by no means so easy to find the poles except for harmonics of the first and second degrees and for particular cases of the higher degrees.

Hence, for many purposes it is desirable to express the harmonic as the sum of a number of other harmonics, each of which has its axes disposed in a symmetrical manner.

Symmetrical System.

132.] The particular forms of harmonics to which it is usual to refer all others are deduced from the general harmonic by placing $i-\sigma$ of the poles at one point, which we shall call the Positive Pole of the sphere, and the remaining $\sigma$ poles at equal distances round one half of the equator.

In this case $\lambda _{1},\ \lambda _{2},\ \dots \lambda _{i-\sigma }$ are each of them equal to $\cos \theta$, and $\lambda _{i-s+1}\dots \lambda _{i}$ are of the form $\sin \theta \cos(\phi -\beta )$. We shall write $\mu$ for $\cos \theta$ and $\nu$ for $\sin \theta$.

Also the value of $\mu _{jj'}$ is unity if $j$ and $j'$ are both less than $i-\sigma$, zero when one is greater and the other less than this quantity, and $\cos n{\frac {\pi }{\sigma }}$ when both are greater.

When all the poles are concentrated at the pole of the sphere, the harmonic becomes a zonal harmonic for which $\sigma =0$. As the zonal harmonic is of great importance we shall reserve for it the symbol $Q_{i}$.

We may obtain its value either from the trigonometrical expression (27), or more directly by differentiation, thus

$Q_{i}=(-1)^{i}{\frac {r^{i+1}}{|{\underline {i}}}}{\frac {d^{i}}{dz^{i}}}\left({\frac {1}{r}}\right),$

(28)

${\begin{array}{c}Q_{i}={\frac {1.3.5\dots (2i-1)}{1.2.3\dots i}}\left\{\mu ^{i}-{\frac {i(i-1)}{2.(2i-1)}}\mu ^{i-2}+{\frac {i(i-1)(i-2)(i-3)}{2.4.(2i-1)(2i-3)}}\mu ^{i-4}-\mathrm {etc} .\right\}\\\\=\sum _{n}\left\{(-1)^{n}{\frac {|{\underline {2i-2n}}}{2^{i}|{\underline {n}}\ |{\underline {i-n}}|{\underline {i-2n}}}}\mu ^{i-2n}\right\}\end{array}}$

(29)

It is often convenient to express $Q_{i}$ as a homogeneous function of $\cos \theta$ and $\sin \theta$, which we shall write $\mu$, and $\nu$ respectively,

${\begin{array}{ll}Q_{i}&=\mu ^{i}-{\frac {i(i-1)}{2.2}}\mu ^{i-2}\nu ^{2}+{\frac {i(i-1)(i-2)(i-3)}{2.2.4.4}}\mu ^{i-4}\nu ^{4}-\mathrm {etc} .\\\\&\sum _{n}\left\{(-1)^{n}{\frac {|{\underline {i}}}{2^{2n}|{\underline {n}}\ |{\underline {n}}\ |{\underline {i-2n}}}}\mu ^{i-2n}\nu ^{2n}\right\}.\end{array}}$

(30)

In this expansion the coefficient of $\mu _{i}$ is unity, and all the other terms involve $\nu$. Hence at the pole, where $\mu =1$ and $\nu =0$, $Q_{i}=1$.

It is shewn in treatises on Laplace’s Coefficients that $Q_{i}$ is the coefficient of $h^{i}$ in the expansion of $\left(1-2\mu h+h^{2}\right)^{-{\frac {1}{2}}}$.

The other harmonics of the symmetrical system are most conveniently obtained by the use of the imaginary coordinates given by Thomson and Tait, Natural Philosophy, vol. i. p. 148,

$\xi =x+{\sqrt {-1}}y,\ \eta =x-{\sqrt {-1}}y.$

(31)

The operation of differentiating with respect to a axes in succession, whose directions make angles ${\tfrac {\pi }{\sigma }}$ with each other in the plane of the equator, may then be written

${\frac {d^{\sigma }}{dh_{1}\dots dh_{\sigma }}}={\frac {d^{\sigma }}{d\xi ^{\sigma }}}+{\frac {d^{\sigma }}{d\eta ^{\sigma }}}.$

(32)

The surface harmonic of degree $i$ and type $\sigma$ is found by differentiating ${\tfrac {1}{r}}$ with respect to $i$ axes, $\sigma$ of which are at equal intervals in the plane of the equator, while the remaining $i-\sigma$ coincide with that of $z$, multiplying the result by $r^{i+1}$ and dividing by $|{\underline {i}}$. Hence

$Y_{i}^{(\sigma )}=(-1)^{i}{\frac {r^{i+1}}{|{\underline {i}}}}{\frac {d^{i-\sigma }}{dz^{i-\sigma }}}\left({\frac {d^{\sigma }}{d\xi ^{\sigma }}}+{\frac {d^{\sigma }}{d\eta ^{\sigma }}}\right)\left({\frac {1}{r}}\right),$

(33)

$=(-1)^{i-s}{\frac {|{\underline {2s}}}{2^{2s}|{\underline {i}}\ |{\underline {s}}}}\left(\xi ^{\sigma }+\eta ^{\sigma }\right)r^{i+1}{\frac {d^{i-\sigma }}{dz^{u-\sigma }}}{\frac {1}{r^{2\sigma +1}}}.$

(34)

Now

$\xi ^{\sigma }+\eta ^{\sigma }=2r^{\sigma }\nu ^{\sigma }\cos(\sigma \phi +\beta ),$

(35)

and

${\frac {d^{i-\sigma }}{dz^{i-\sigma }}}{\frac {1}{r^{2\sigma +1}}}=(-1)^{i-\sigma }{\frac {|{\underline {i+\sigma }}}{2\sigma }}{\frac {1}{r^{i+\sigma +1}}}\vartheta _{i}^{(\sigma )}.$

(36)

Hence

$Y_{i}^{(\sigma )}=2{\frac {|{\underline {i+\sigma }}}{2^{2\sigma }|{\underline {i}}\ |{\underline {\sigma }}}}\vartheta _{i}^{(\sigma )}\cos(\sigma \phi +\beta ),$

(37)

where the factor 2 must be omitted when $\sigma =0$.

The quantity $\vartheta _{i}^{(\sigma )}$ is a function of $\theta$, the value of which is given in Thomson and Tait’s Natural Philosophy, vol. i. p. 149.

It may be derived from $Q_{i}$ by the equation

$\vartheta _{i}^{(\sigma )}=2^{\sigma }{\frac {|{\underline {i-\sigma }}\ |{\underline {\sigma }}}{|{\underline {i+\sigma }}}}\nu ^{\sigma }{\frac {d^{\sigma }}{d\mu ^{\sigma }}}Q_{i},$

(38)

where $Q_{i}$ is expressed as a function of $\mu$ only.

Performing the differentiations on $Q_{i}$ as given in equation (29), we obtain

$\vartheta _{i}^{(\sigma )}=\nu ^{\sigma }\sum \left\{(-1)^{n}{\frac {|{\underline {i-\sigma }}\ |{\underline {\sigma }}\ |{\underline {2i-2n}}}{2^{i-\sigma }|{\underline {i+\sigma }}\ |{\underline {n}}\ |{\underline {i-n}}\ |{\underline {i-\sigma -2n}}}}\mu ^{i-\sigma -2n}\right\}.$

(39)

We may also express it as a homogeneous function of $\mu$ and $\nu$,

$\vartheta _{i}^{(\sigma )}=\nu ^{\sigma }\sum \left\{(-1)^{n}{\frac {|{\underline {i-\sigma }}\ |{\underline {\sigma }}}{2^{2\sigma }|{\underline {n}}\ |{\underline {\sigma +n}}\ |{\underline {i-\sigma -2n}}}}\mu ^{i-\sigma -2n}\nu ^{2n}\right\}.$

(40)

In this expression the coefficient of the first term is unity, and the others may be written down in order by the application of Laplace’s equation.

The following relations will be found useful in Electrodynamics. They may be deduced at once from the expansion of $Q_{i}$.

$\mu Q_{i}-Q_{i+1}={\frac {1}{i+1}}\nu ^{2}{\frac {dQ_{i}}{d\mu }}={\frac {i}{2}}\nu \vartheta _{i}^{1},$

(41)

$Q_{i-1}-\mu Q_{i}={\frac {1}{i}}\nu ^{2}{\frac {dQ_{i}}{d\mu }}={\frac {i+1}{2}}\nu \vartheta _{i}^{1}.$

(42)

On Harmonics of Positive Degree.

133.] We have hitherto considered the spherical surface harmonic $Y_{i}$ as derived from the solid harmonic

$V_{i}=|{\underline {i}}\ M_{i}{\frac {Y_{i}}{r^{i+1}}}.$

This solid harmonic is a homogeneous function of the coordinates of the negative degree $-(i+1)$. Its values vanish at an infinite distance and become infinite at the origin.

We shall now shew that to every such function there corresponds another which vanishes at the origin and has infinite values at an infinite distance, and is the corresponding solid harmonic of positive degree $i$.

A solid harmonic in general may be defined as a homogeneous function of x, y, and z, which satisfies Laplace’s equation

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}=0.$

Let $H_{i}$ be a homogeneous function of the degree $i$, such that

$H_{1}|{\underline {i}}\ M_{i}r^{i}Y_{i}=r^{2i+1}V_{i}.$

(43)

Then

${\frac {dH_{i}}{dx}}=(2i+1)r^{2i-1}xV_{i}+r^{2i+1}{\frac {dV_{i}}{dx}},$ ${\frac {d^{2}H_{i}}{dx^{2}}}=(2i+1)\left((2i-1)x^{2}+r^{2}\right)r^{2i-3}V_{i}+2(2i+1)r^{2i-1}x{\frac {dV_{i}}{dx}}+r^{2i+1}{\frac {d^{2}V_{i}}{dx^{2}}}.$

Hence

${\begin{array}{l}{\frac {d^{2}H_{i}}{dx^{2}}}+{\frac {d^{2}H_{i}}{dy^{2}}}+{\frac {d^{2}H_{i}}{dz^{2}}}=(2i+1)(2i+2)r^{2i-1}V_{i}\\\\+2(2i+1)r^{2i-1}\left(x{\frac {dV_{i}}{dx}}+y{\frac {dV_{i}}{dy}}+z{\frac {dV_{i}}{dz}}\right)+r^{2i+1}\left({\frac {d^{2}V_{i}}{dx^{2}}}+{\frac {d^{2}V_{i}}{dy^{2}}}+{\frac {d^{2}V_{i}}{dz^{2}}}\right).\end{array}}$

(44)

Now, since $V_{i}$ is a homogeneous function of negative degree $i+1$,

$x{\frac {dV_{i}}{dx}}+y{\frac {dV_{i}}{dy}}+z{\frac {dV_{i}}{dz}}=-(i+1)V_{i}.$

(45)

The first two terms therefore of the right hand member of equation (44) destroy each other, and, since $V_{i}$ satisfies Laplace’s equation, the third term is zero, so that $H_{i}$ also satisfies Laplace’s equation, and is therefore a solid harmonic of degree $i$.

We shall next shew that the value of $H_{i}$ thus derived from $V_{i}$ is of the most general form.

A homogeneous function of $x,y,z$ of degree $i$ contains

${\frac {1}{2}}(i+1)(i+2)$

terms. But

$-\nabla ^{2}H_{i}={\frac {d^{2}H_{i}}{dx^{2}}}+{\frac {d^{2}H_{i}}{dy^{2}}}+{\frac {d^{2}H_{i}}{dz^{2}}}$

is a homogeneous function of degree $i-2$, and therefore contains ${\tfrac {1}{2}}i(i-1)$ terms, and the condition $\nabla ^{2}H_{i}=0$ requires that each of these must vanish. There are therefore ${\tfrac {1}{2}}i(i-1)$ equations between the coefficients of the ${\tfrac {1}{2}}(i+1)(i+2)$ terms of the homogeneous function, leaving $2i+1$ independent constants in the most general form of $H_{i}$.

But we have seen that $V_{i}$ has $2i+1$ independent constants, therefore the value of $H_{i}$ is of the most general form.

Application of Solid Harmonics to the Theory of Electrified Spheres.

134.] The function $V_{i}$ satisfies the condition of vanishing at infinity, but does not satisfy the condition of being everywhere finite, for it becomes infinite at the origin.

The function $H_{i}$ satisfies the condition of being finite and continuous at finite distances from the origin, but does not satisfy the condition of vanishing at an infinite distance.

But if we determine a closed surface from the equation

$V_{i}=H_{i},$

(46)

and make $H_{i}$ the potential function within the closed surface and $V_{i}$ the potential outside it, then by making the surface-density $\sigma$ satisfy the characteristic equation

${\frac {dH_{i}}{dr}}-{\frac {dV_{i}}{dr}}+4\pi \sigma =0,$

(47)

we shall have a distribution of potential which satisfies all the conditions.

It is manifest that if $H_{i}$ and $V_{i}$ are derived from the same value of $Y_{i}$, the surface $H_{i}=V_{i}$ will be a spherical surface, and the surface-density will also be derived from the same value of $Y_{i}$.

Let $a$ be the radius of the sphere, and let

$H_{i}=Ar^{i}Y_{i},\ V_{i}=B{\frac {Y_{i}}{r^{i+1}}},\ \sigma =CY_{i}.$

(48)

Then at the surface of the sphere, where $r=a$,

$Aa^{i}={\frac {B}{a^{i+1}}},$

and

${\frac {dV}{dr}}-{\frac {dH}{dr}}=-4\pi \sigma ;$

or

$(i+1){\frac {B}{a^{i+2}}}+ia^{i-1}A=4\pi C;$

whence we find $H_{i}$ and $V_{i}$ in terms of $C$,

$H_{i}={\frac {4\pi C}{2i+1}}{\frac {r^{i}}{a^{i-1}}}Y_{i},\ V_{i}={\frac {4\pi C}{2i+1}}{\frac {a^{i+2}}{r^{i+1}}}Y_{i}.$

(49)

We have now obtained an electrified system in which the potential is everywhere finite and continuous. This system consists of a spherical surface of radius $a$, electrified so that the surface-density is everywhere $CY_{i}$, where $C$ is some constant density and $Y_{i}$ is a surface harmonic of degree $i$. The potential inside this sphere, arising from this electrification, is everywhere $H_{i}$, and the potential outside the sphere is $V_{i}$.

These values of the potential within and without the sphere might have been obtained in any given case by direct integration, but the labour would have been great and the result applicable only to the particular case.

135.] We shall next consider the action between a spherical surface, rigidly electrified according to a spherical harmonic, and an external electrified system which we shall call $E$.

Let $V$ be the potential at any point due to the system $E$, and $V_{i}$ that due to the spherical surface whose surface-density is $\sigma$.

Then, by Green’s theorem, the potential energy of $E$ on the electrified surface is equal to that of the electrified surface on $E$, or

$\iint V\sigma dS=\sum V_{i}dE,$

(50)

where the first integration is to be extended over every element $dS$ of the surface of the sphere, and the summation $\sum$ is to be extended to every part $dE$ of which the electrified system $E$ is composed.

But the same potential function $V_{i}$ may be produced by means of a combination of $2^{i}$ electrified points in the manner already described. Let us therefore find the potential energy of $E$ on such a compound point.

If $M_{0}$ is the charge of a single point of degree zero, then $M_{0}V$ is the potential energy of $V$ on that point.

If there are two such points, a positive and a negative one, at the positive and negative ends of a line $h_{i}$, then the potential energy of $E$ on the double point will be

$-M_{0}V+M_{0}\left(V+h_{1}{\frac {dV}{dh_{1}}}+{\frac {1}{2}}h^{2}{\frac {d^{2}V}{dh_{1}^{2}}}+\mathrm {etc} .\right);$

and when $M_{0}$ increases and $h_{1}$ diminishes indefinitely, but so that

$M_{0}h_{1}=M_{1},$

the value of the potential energy will be for a point of the first degree

$M_{1}{\frac {dV}{dh_{1}}}.$

Similarly for a point of degree $i$ the potential energy with respect to $E$ will be

$M_{1}{\frac {d^{i}V}{dh_{1}dh_{2}\dots dh_{i}}}$

This is the value of the potential energy of $E$ upon the singular point of degree $i$. That of the singular point on $E$ is $\sum V_{i}dE$ and, by Green’s theorem, these are equal. Hence, by equation (50),

$\iint V\sigma \ dS=M_{i}{\frac {d^{i}V}{dh_{1}\dots dh_{i}}}.$

If $\sigma =CY_{i}$ where $C$ is a constant quantity, then, by equations (49) and (14),

$M_{i}={\frac {4\pi C}{|{\underline {i}}}}{\frac {a^{i+2}}{2i+1}}$

(51)

Hence, if $V$ is any potential function whatever which satisfies Laplace’s equation within the spherical surface of radius $a$, then the integral of $VY_{i}dS$, extended over every element $dS_{1}$ of the surface of a sphere of radius $a$, is given by the equation

$\iint VY_{i}dS={\frac {4\pi }{|{\underline {i}}}}{\frac {a^{i+2}}{2i+1}}{\frac {d^{i}V}{dh_{1}\dots dh_{1}}};$

(52)

where the differentiations of $V$ are taken with respect to the axes of the harmonic $Y_{i}$, and the value of the differential coefficient is that at the centre of the sphere.

136.] Let us now suppose that $V$ is a solid harmonic of positive degree $j$ of the form j

$V={\frac {r^{j}}{a^{j}}}Y_{j}.$

(53)

At the spherical surface, $r=a$, the value of $V$ is the surface harmonic $Y_{j}$, and equation (52) becomes

$\iint Y_{i}Y_{j}dS={\frac {4\pi }{|{\underline {i}}}}{\frac {a^{i-j+2}}{2i+1}}{\frac {d^{i}\left(r^{j}Y_{j}\right)}{dh_{1}\dots dh_{i}}},$

(54)

where the value of the differential coefficient is that at the centre of the sphere.

When $i$ is numerically different from $j$, the surface-integral of the product $Y_{i}Y_{j}$ vanishes. For, when $i$ is less than $j$, the result of the differentiation in the second member of (54) is a homogeneous function of x, y, and z, of degree $j-i$, the value of which at the centre of the sphere is zero. If $i$ is equal to $j$ the result is a constant, the value of which will be determined in the next article. If the differentiation is carried further, the result is zero. Hence the surface-integral vanishes when $i$ is greater than $j$.

137.] The most important case is that in which the harmonic $r^{j}Y_{j}$ is differentiated with respect to $i$ new axes in succession, the numerical value of $j$ being the same as that of $i$, but the directions of the axes being in general different. The final result in this case is a constant quantity, each term being the product of $i$ cosines of angles between the different axes taken in pairs. The general form of such a product may be written symbolically

$\mu _{ii}^{s}\mu _{jj}^{s}\mu _{ij}^{i-2s},$

which indicates that there are $s$ cosines of angles between pairs of axes of the first system and $s$ between axes of the second system, the remaining $i-2s$ cosines being between axes one of which belongs to the first and the other to the second system.

In each product the suffix of every one of the $2i$ axes occurs once, and once only.

The number of different products for a given value of $s$ is

$N={\frac {\left(|{\underline {i}}\right)^{2}}{2^{2s}\left(|{\underline {s}}\right)^{2}|{\underline {i-2s}}}}.$

(55)

The final result is easily obtained by the successive differentiation of

$r_{j}Y_{j}={\frac {1}{|{\underline {j}}}}S\left\{(-1)^{s}{\frac {|{\underline {2j-2s}}}{2^{j-s}|{\underline {j-s}}}}r^{2s}\sum \left(p^{j-2s}\mu ^{s}\right)\right\}.$

Differentiating this $i$ times in succession with respect to the new axes, so as to obtain any given combination of the axes in pairs, we find that in differentiating $r^{2s}$ with respect to $s$ of the new axes, which are to be combined with other axes of the new system, we introduce the numerical factor $2s(2s-2)\dots 2$, or $2^{s}|{\underline {s}}$. In continuing the differentiation the $p$’s become converted into $\mu$’s, but no numerical factor is introduced. Hence

${\frac {d^{i}}{dh_{1}\dots dh_{i}}}r^{j}Y_{j}={\frac {1}{|{\underline {i}}}}S\left\{(-1)^{s}{\frac {|{\underline {2i-2s}}\ |{\underline {s}}}{2^{i-s}|{\underline {i-s}}}}\sum \left(\mu _{ii}^{s}\mu _{jj}^{s}\mu _{ij}^{i-2s}\right)\right\}$

(56)

Substituting this result in equation (54) we find for the value of the surface-integral of the product of two surface harmonics of the same degree, taken over the surface of a sphere of radius $a$,

$\iint Y_{i}Y_{j}dS={\frac {4\pi a^{2}}{(2i+1)\left(|{\underline {i}}\right)^{2}}}S\left\{(-1)^{s}{\frac {|{\underline {2i-2s}}\ |{\underline {s}}}{2^{i-s}|{\underline {i-s}}}}\sum \left(\mu _{ii}^{s}\mu _{jj}^{s}\mu _{ij}^{i-2s}\right)\right\}$

(57)

This quantity differs from zero only when the two harmonics are of the same degree, and even in this case, when the distribution of the axes of the one system bears a certain relation to the distribution of the axes of the other, this integral vanishes. In this case, the two harmonics are said to be conjugate to each other.

On Conjugate Harmonics.

138.] If one harmonic is given, the condition that a second harmonic of the same degree may be conjugate to it is expressed by equating the right hand side of equation (57) to zero.

If a third harmonic is to be found conjugate to both of these there will be two equations which must be satisfied by its $2i$ variables.

If we go on constructing new harmonics, each of which is conjugate to all the former harmonics, the variables will be continually more and more restricted, till at last the $(2i+1)$th harmonic will have all its variables determined by the $2i$ equations, which must be satisfied in order that it may be conjugate to the $2i$ preceding harmonics.

Hence a system of $2i+1$ harmonics of degree $i$ may be constructed, each of which is conjugate to all the rest. Any other harmonic of the same degree may be expressed as the sum of this system of conjugate harmonics each multiplied by a coefficient.

The system described in Art. 132, consisting of $2i+1$ harmonics symmetrical about a single axis, of which the first is zonal, the next $i-1$ pairs tesseral, and the last pair sectorial, is a particular case of a system of $2i+1$ harmonics, all of which are conjugate to each other. Sir W. Thomson has shewn how to express the conditions that $2i+1$ perfectly general harmonics, each of which, however, is expressed as a linear function of the $2i+1$ harmonics of this symmetrical system, may be conjugate to each other. These conditions consist of $i(2i+1)$ linear equations connecting the $(2i+1)^{2}$ coefficients which enter into the expressions of the general harmonics in terms of the symmetrical harmonics.

Professor Clifford has also shewn how to form a conjugate system of $2i+1$ sectorial harmonics having different poles.

Both these results were communicated to the British Association in 1871.

139.] If we take for $Y_{j}$ the zonal harmonic $Q_{j}$, we obtain a remarkable form of equation (57).

In this case all the axes of the second system coincide with each other.

The cosines of the form $\mu _{ij}$, will assume the form $\lambda$ where $\lambda$ is the cosine of the angle between the common axis of $Q_{j}$ and an axis of the first system.

The cosines of the form $\mu _{ij}$ will all become equal to unity.

The number of combinations of $s$ symbols, each of which is distinguished by two out of $i$ suffixes, no suffix being repeated, is

$N={\frac {|{\underline {i}}}{2^{s}|{\underline {s}}\ |{\underline {i-2s}}}};$

(58)

and when each combination is equal to unity this number represents the sum of the products of the cosines $\mu _{jj}$, or $\sum \left(\mu _{jj}^{s}\right)$.

The number of permutations of the remaining $i-2s$ symbols of the second set of axes taken all together is $|{\underline {i-2s}}$. Hence

$\sum \left(\mu _{ij}^{i+2s}\right)=|{\underline {i-2s}}\sum \lambda ^{i-2s}.$

Equation (57) therefore becomes, when $Y_{j}$ is the zonal harmonic,

${\begin{array}{ll}\iint Y_{i}Q_{j}dS&={\frac {4\pi a^{2}}{(2i+1)|{\underline {i}}}}S\left\{(-1)^{s}{\frac {|{\underline {2i-2s}}}{2^{i-s}|{\underline {i-s}}}}\sum \left(\lambda ^{i-2s}\mu ^{s}\right)\right\},\\\\&={\frac {4\pi a^{2}}{2i+1}}Y_{i(j)},\end{array}}$

(60)

where $Y_{i(j)}$ denotes the value of $Y_{i}$ in equation (27) at the common pole of all the axes of $Q_{j}$.

140.] This result is a very important one in the theory of spherical harmonics, as it leads to the determination of the form of a series of spherical harmonics, which expresses a function having any arbitrarily assigned value at each point of a spherical surface.

For let $F$ be the value of the function at any given point of the sphere, say at the centre of gravity of the element of surface $dS$, and let $Q_{i}$ be the zonal harmonic of degree $i$ whose pole is the point $P$ on the sphere, then the surface-integral

$\iint FQ_{i}dS$

extended over the spherical surface will be a spherical harmonic of degree $i$, because it is the sum of a number of zonal harmonics whose poles are the various elements $dS$, each being multiplied by $FdS$. Hence, if we make

$A_{i}Y_{i}={\frac {2i+1}{4\pi a^{2}}}\iint FQ_{i}dS,$

(61)

we may expand F in the form

$F=A_{0}Y_{0}+A_{1}Y_{1}+\mathrm {etc} .+A_{i}Y_{i},$

(62)

or

$F={\frac {1}{4\pi a^{2}}}\left\{\iint FQ_{0}dS+3\iint FQ_{1}dS+\mathrm {etc} .+(2i+1)\iint fQ_{i}dS\right\}.$

(63)

This is the celebrated formula of Laplace for the expansion in a series of spherical harmonics of any quantity distributed over the surface of a sphere. In making use of it we are supposed to take a certain point $P$ on the sphere as the pole of the zonal harmonic $Q_{i}$, and to find the surface-integral

$\iint FQ_{i}dS$

over the whole surface of the sphere. The result of this operation when multiplied by $2i+1$ gives the value of $A_{i}Y_{i}$ at the point $P$, and by making $P$ travel over the surface of the sphere the value of $A_{i}Y_{i}$ at any other point may be found.

But $A_{i}Y_{i}$ is a general surface harmonic of degree $i$, and we wish to break it up into the sum of a series of multiples of the $2i+1$ conjugate harmonics of that degree.

Let $P_{i}$ be one of these conjugate harmonics of a particular type, and let $B_{i}P_{i}$ be the part of $A_{i}Y_{i}$ belonging to this type.

We must first find

$M=\iint P_{i}P_{i}dS,$

(64)

which may be done by means of equation (57), making the second set of poles the same, each to each, as the first set.

We may then find the coefficient $B_{i}$ from the equation

$B_{i}={\frac {1}{M}}\iint FP_{i}dS.$

For suppose $F$ expanded in terms of spherical harmonics, and let $B_{j}P_{j}$ be any term of this expansion. Then, if the degree of $P_{j}$ is different from that of $P_{i}$, or if, the degree being the same, $P_{j}$ is conjugate to $P_{i}$, the result of the surface-integration is zero. Hence the result of the surface-integration is to select the coefficient of the harmonic of the same type as $P_{i}$.

The most remarkable example of the actual development of a function in a series of spherical harmonics is the calculation by Gauss of the harmonics of the first four degrees in the expansion of the magnetic potential of the earth, as deduced from observations in various parts of the world.

He has determined the twenty-four coefficients of the three conjugate harmonics of the first degree, the five of the second, seven of the third, and nine of the fourth, all of the symmetrical system. The method of calculation is given in his General Theory of Terrestrial Magnetism.

141.] When the harmonic $P_{i}$ belongs to the symmetrical system we may determine the surface-integral of its square extended over the sphere by the following method.

The value of $r^{i}Y_{i}^{\sigma }$ is, by equations (34) and (36),

$r^{i}Y_{i}^{(\sigma )}={\frac {|{\underline {i}}+\sigma }{2^{\sigma }|{\underline {i}}\ |{\underline {\sigma }}}}\left(\xi ^{\sigma }+\eta ^{\sigma }\right)\left(z^{i-\sigma }-{\frac {(i-\sigma )(i-\sigma -1)}{4(\sigma +1)}}z^{i-\sigma -2}\xi \eta +\mathrm {etc} .\right);$

and by equations (33) and (54),

$\iint \left(Y_{i}^{(\sigma )}\right)^{2}dS={\frac {4\pi }{|{\underline {i}}}}{\frac {a^{2}}{2i+1}}\cdot {\frac {d^{i-\sigma }}{dz^{i-\sigma }}}\left({\frac {d^{\sigma }}{d\xi ^{\sigma }}}+{\frac {d^{\sigma }}{d\eta ^{\sigma }}}\right)\left(r_{i}Y_{i}^{(\sigma )}\right)$

Performing the differentiations, we find that the only terms which do not disappear are those which contain $z^{i-\sigma }$. Hence

$\iint \left(Y_{i}^{(\sigma )}\right)^{2}dS={\frac {8\pi a^{2}}{2i+1}}\cdot {\frac {|{\underline {i+\sigma }}\ |{\underline {i-\sigma }}}{2^{2\sigma }|{\underline {i}}\ |{\underline {i}}}},$

(66)

except when $\sigma =0$, in which case we have, by equation (60),

$\iint \left(Q_{i}\right)^{2}dS={\frac {8\pi a^{2}}{2i+1}}.$

These expressions give the value of the surface-integral of the square of any surface harmonic of the symmetrical system.

We may deduce from this the value of the integral of the square of the function $\vartheta _{i}^{(\sigma )}$, given in Art. 132,

$\int _{-1}^{+1}\left(\vartheta _{i}^{(\sigma )}\right)^{2}d\mu ={\frac {2}{2i+1}}{\frac {2^{2\sigma }|{\underline {i-\sigma }}\left(|{\underline {\sigma }}\right)^{2}}{|{\underline {i+\sigma }}}}.$

(68)

This value is identical with that given by Thomson and Tait, and is true without exception for the case in which $\sigma =0$.

142.] The spherical harmonics which I have described are those of integral degrees. To enter on the consideration of harmonics of fractional, irrational, or impossible degrees is beyond my purpose, which is to give as clear an idea as I can of what these harmonics are. I have done so by referring the harmonic, not to a system of polar coordinates of latitude and longitude, or to Cartesian coordinates, but to a number of points on the sphere, which I have called the Poles of the harmonic. Whatever be the type of a harmonic of the degree $i$, it is always mathematically possible to find $i$ points on the sphere which are its poles. The actual calculation of the position of these poles would in general involve the solution of a system of $2i$ equations of the degree $i$. The conception of the general harmonic, with its poles placed in any manner on the sphere, is useful rather in fixing our ideas than in making calculations. For the latter purpose it is more convenient to consider the harmonic as the sum of $2i+1$ conjugate harmonics of selected types, and the ordinary symmetrical system, in which polar coordinates are used, is the most convenient. In this system the first type is the zonal harmonic $Q_{i}$, in which all the axes coincide with the axis of polar coordinates. The second type is that in which $i-1$ of the poles of the harmonic coincide at the pole of the sphere, and the remaining one is on the equator at the origin of longitude. In the third type the remaining pole is at 90° of longitude.

In the same way the type in which $i-\sigma$ poles coincide at the pole of the sphere, and the remaining a are placed with their axes at equal intervals ${\tfrac {\pi }{\sigma }}$ round the equator, is the type $2\sigma$, if one of the poles is at the origin of longitude, or the type $2\sigma +1$ if it is at longitude ${\tfrac {\pi }{2\sigma }}$.

143.] It appears from equation (60) that it is always possible to express a harmonic as the sum of a system of zonal harmonics of the same degree, 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 degrees, 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 sum of two zonal harmonics of the third degree whose axes are inclined 120° in the plane of the paper, and the sum is the harmonic of the second type in which $\sigma =1$, the axis being perpendicular to the paper.

Fig. VI

In Fig. VII the harmonic is also of the third degree, but the axes of the zonal harmonics of which it is the sum are inclined 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. VII

Fig. VIII represents the difference of two zonal harmonics of the fourth degree whose axes are at right angles. The result is a tesseral harmonic for which $i=4$, $\sigma =2$.

Fig. VIII

Fig. IX represents the sum of the same zonal harmonics. The result gives some notion of one type of the more general harmonic of the fourth degree. 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 connected part of the spherical surface which lies outside the ovals, the harmonic is negative.

Fig. IX

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 degree.

Fig. V

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 magnetism if it were distributed according to the most simple type.

144.] It appears from equation (52), by making $i=0$, that if $V$ satisfies Laplace’s equation throughout the space occupied by a sphere of radius $a$, then the integral

$\iint V\ dS=4\pi a^{2}V_{0},$

(69)

where the integral is taken over the surface of the sphere, $dS$ being an element of that surface, and $V_{0}$ is the value of $V$ at the centre of the sphere. This theorem may be thus expressed.

The value of the potential at the centre of a sphere is the mean value of the potential for all points of its surface, provided the potential be due to an electrified system, no part of which is within the sphere.

It follows from this that if $V$ satisfies Laplace’s equation through out a certain continuous region of space, and if, throughout a finite portion, however small, of that space, $V$ is constant, it will be constant throughout the whole continuous region.

If not, let the space throughout which the potential has a constant value $C$ be separated by a surface $S$ from the rest of the region in which its values differ from $C$, then it will always be possible to find a finite portion of space touching $S$ and out side of it in which $V$ is either everywhere greater or everywhere less than $C$.

Now describe a sphere with its centre within $S$, and with part of its surface outside $S$, but in a region throughout which the value of $V$ is every where greater or everywhere less than $C$.

Then the mean value of the potential over the surface of the sphere will be greater than its value at the centre in the first case and less in the second, and therefore Laplace’s equation cannot be satisfied throughout the space occupied by the sphere, contrary to our hypothesis. It follows from this that if $V=C$ throughout any portion of a connected region, $V=C$ throughout the whole of the region which can be reached in any way by a body of finite size without passing through electrified matter. (We suppose the body to be of finite size because a region in which $V$ is constant may be separated from another region in which it is variable by an electrified surface, certain points or lines of which are not electrified, so that a mere point might pass out of the region through one of these points or lines without passing through electrified matter.) This remarkable theorem is due to Gauss. See Thomson and Tait’s Natural Philosophy, § 497.

It may be shewn in the same way that if throughout any finite portion of space the potential has a value which can be expressed by a continuous mathematical formula satisfying Laplace’s equation, the potential will be expressed by the same formula throughout every part of space which can be reached without passing through electrified matter.

For if in any part of this space the value of the function is $V'$, different from $V$, that given by the mathematical formula, then, since both $V$ and $V'$ satisfy Laplace’s equation, $U=V'-V$ does. But within a finite portion of the space $U=0$, therefore by what we have proved $U=0$ throughout the whole space, or $V'=V$.

145.] Let $Y_{i}$ be a spherical harmonic of $i$ degrees and of any type. Let any line be taken as the axis of the sphere, and let the harmonic be turned into $n$ positions round the axis, the angular distance between consecutive positions being ${\tfrac {2\pi }{n}}$.

If we take the sum of the $n$ harmonics thus formed the result will be a harmonic of $i$ degrees, which is a function of $\theta$ and of the sines and cosines of $n\phi$.

If $n$ is less than $i$ the result will be compounded of harmonics for which $s$ is zero or a multiple of $n$ less than $i$, but if $n$ is greater than $i$ the result is a zonal harmonic. Hence the following theorem :

Let any point be taken on the general harmonic $Y_{i}$, and let a small circle be described with this point for centre and radius $\theta$, and let $n$ points be taken at equal distances round this circle, then if $Q_{i}$ is the value of the zonal harmonic for an angle $\theta$, and if $Y_{i}'$ is the value of $Y_{i}$ at the centre of the circle, then the mean of the $n$ values of $Y_{i}$ round the circle is equal to $Q_{i}Y_{i}'$ provided $n$ is greater than $i$.

If $n$ is greater than $i+s$, and if the value of the harmonic at each point of the circle be multiplied by $\sin s\phi$ or $\cos s\phi$ where $s$ is less than $i$, and the arithmetical mean of these products be A_$s$, then if $\vartheta _{i}^{\prime (s)}$ is the value of $\vartheta _{i}^{(s)}$ for the angle $\theta$, the coefficient of $\sin s\phi$ or $\cos s\phi$ in the expansion of $Y_{i}$ will be

$2A{\frac {\vartheta _{i}^{(s)}}{\vartheta _{i}^{\prime (s)}}}$

In this way we may analyse Y_i into its component conjugate harmonics by means of a finite number of ascertained values at selected points on the sphere.

Application of Spherical Harmonic Analysis to the Determination of the Distribution of Electricity on Spherical and nearly Spherical Conductors under the Action of known External Electrical Forces.

146.] We shall suppose that every part of the electrified system which acts on the conductor is at a greater distance from the centre of the conductor than the most distant part of the conductor itself, or, if the conductor is spherical, than the radius of the sphere.

Then the potential of the external system, at points within this distance, may be expanded in a series of solid harmonics of positive degree

$V=A_{0}+A_{1}rY_{1}+\mathrm {etc} .+A_{i}Y_{i}r^{i}.$

(70)

The potential due to the conductor at points outside it may be expanded in a series of solid harmonics of the same type, but of negative degree

$U=B_{0}{\frac {1}{r}}+B_{1}Y_{1}{\frac {1}{r^{2}}}+\mathrm {etc} .+B_{i}Y_{i}{\frac {1}{r^{i+1}}}$

(71)

At the surface of the conductor the potential is constant and equal, say, to $C$. Let us first suppose the conductor spherical and of radius $a$. Then putting $r=a$, we have $U+V=C$, or, equating the coefficients of the different degrees,

${\begin{array}{l}B_{0}=a\left(C-A_{0}\right),\\B_{1}=-a^{3}A_{1},\\-\ -\ -\ -\ -\\B_{i}=-a^{2i+1}A_{i}.\end{array}}$

(72)

The total charge of electricity on the conductor is $B_{0}$.

The surface-density at any point of the sphere may be found from the equation

${\begin{array}{ll}4\pi \sigma &={\frac {dV}{dr}}-{\frac {dU}{dr}}\\\\&={\frac {B_{0}}{a^{2}}}-3a^{3}A_{1}rY_{1}-\mathrm {etc} .-(2i+1)a^{2i+1}A_{i}Y_{i}.\end{array}}$

(73)

Distribution of Electricity on a nearly Spherical Conductor.

Let the equation of the surface of the conductor be

$r=a(1+F)$

where $F$ is a function of the direction of $r$, and is a numerical quantity the square of which may be neglected.

Let the potential due to the external electrified system be expressed, as before, in a series of solid harmonics of positive degree, and let the potential $U$ be a series of solid harmonics of negative degree. Then the potential at the surface of the conductor is obtained by substituting the value of $r$ from equation (74) in these series.

Hence, if $C$ is the value of the potential of the conductor and $B_{0}$ the charge upon it,

${\begin{array}{ll}C=&A_{0}+A_{1}aY_{1}+\dots +A_{i}a^{i}Y_{i},\\\\&\qquad +A_{1}aFY_{1}+\dots +iA_{i}a^{i}FY_{i},\\\\&+B_{0}{\frac {1}{a}}+B_{1}{\frac {1}{a^{2}}}Y_{1}+\dots +B_{i}a^{-(i+1)}Y_{i}+\dots +B_{j}a^{-j+1}Y_{j},\\\\&-B_{0}{\frac {1}{a}}-2B_{1}{\frac {1}{a^{2}}}FY_{1}+\dots -(i+1)B_{i}a^{-(i+1)}FY_{i}+\dots -(j+1)B_{j}a^{-(j+1)}FY_{j}.\end{array}}$

(75)

Since $F$ is very small compared with unity, we have first a set of equations of the form (72), with the additional equation

${\begin{array}{ll}0=&-B_{0}{\frac {1}{a}}F+3A_{1}aFY_{1}+\mathrm {etc} .+(i+1)A_{i}a^{i}FY_{i}\\\\&+\sum \left(B_{j}a^{-(j+1)}Y_{j}\right)-\sum \left((j+1)B_{j}a^{-(j+1)}FY_{j}\right).\end{array}}$

(76)

To solve this equation we must expand $F$, $FY_{1}\dots FY_{i}$ in terms of spherical harmonics. If $F$ can be expanded in terms of spherical harmonics of degrees lower than $k$, then $FY_{i}$ can be expanded in spherical harmonics of degrees lower than $i+k$.

Let therefore

$B_{0}{\frac {1}{a}}F-3A_{1}aFY_{1}-\dots -(2i+1)A_{i}a^{i}FY_{i}=\sum \left(B_{j}a^{-(j+1)}Y_{j}\right),$

(77)

then the coefficients $B_{j}$ will each of them be small compared with the coefficients $B_{0}\dots B_{i}$ on account of the smallness of $F$, and therefore the last term of equation (76), consisting of terms in $B_{j}F$, may be neglected.

Hence the coefficients of the form $B_{j}$ may be found by expanding equation (76) in spherical harmonics.

For example, let the body have a charge $B_{0}$, and be acted on by no external force.

Let $F$ be expanded in a series of the form

$F=S_{1}Y_{1}+\mathrm {etc} .+S_{k}Y_{k}.$

(78)

Then

$B_{0}{\frac {1}{a}}S_{1}Y_{1}+\mathrm {etc} .+B_{0}{\frac {1}{a}}S_{k}Y_{k}=\sum \left(B_{j}a^{-(j+1)}Y_{j}\right),$

(79)

or the potential at any point outside the body is

$U={\frac {1}{a}}B_{0}\left({\frac {a}{r}}+{\frac {a^{2}}{r^{2}}}S_{1}Y_{1}+\dots +{\frac {a^{k+1}}{r^{k+1}}}S_{k}Y_{k}\right);$

(80)

and if $\sigma$ is the surface-density at any point

$4\pi \sigma =-{\frac {dU}{dr}},$

or

$4\pi a\sigma =B_{0}\left(1+S_{2}Y_{2}+\dots +(k-1)S_{k}Y_{k}\right).$

(81)

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 degree $k$, the ratio of the difference of the superficial densities at any two points to their sum will be $k-1$ times the ratio of the difference of the radii of the same two points to their sum.

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1. Gauss. Werke, bd. v. s. 361.