Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/219

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

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