different ways, and if we do so to all the terms, we shall obtain the whole permutations of symbols, the number of which is . Let the sum of all terms of this kind be written in the abbreviated form
If we wish to express that a particular symbol occurs among the ’s only, or among the ’s only, we write it as a suffix to the or the . Thus the equation
expresses that the whole system of terms may be divided into two portions, in one of which the symbol 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
This is evidently true when and when . We shall shew that if it is true for it is true for . We may write the series
where indicates a summation in which all values of not greater than are to be taken.
Multiplying by , and remembering that , we obtain by (14), for the value of the solid harmonic of negative degree, and moment unity,
Differentiating with respect to a new axis whose symbol is , we should obtain with its sign reversed,
If we wish to obtain the terms containing cosines with double suffixes we must diminish by unity in the second term, and we find
If we now make
and this value of is the same as that obtained by changing