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

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

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 $$i$$

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

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,

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

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

If we now make

then

and this value of $$V_{i+1}$$ is the same as that obtained by changing $$i$$