Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/203

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 $i$