Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/204

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

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