OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
381 
wlier ej=^/ —1, g=x J r jy, y—x—jy. It will be observed that (8) is not in accordance 
with the general definition (7), for if we put <x=0 we get (i, o) = 2P*. 
As regards the name of Spherical Harmonic introduced by Thomson and Tait, it 
seems desirable that the names of Legendre and Laplace should be in some way 
retained in connexion with the different kinds of harmonics with which they are asso¬ 
ciated. From the point of view of the former writers there is perhaps some difficulty 
in doing this. We shall however continue to give the name of Laplace’s coefficient 
to what Thomson and Tait call the Biaxal Harmonic or the formula which expresses Q; 
in terms of the harmonics referred to fixed axes of reference. With the definitions 
given above, this is 
Q;=P/P/-f, &C., 
2 2 °—h! i! 
'i + cr ■! i — a\ 
{(i, cr)(i, 
o-)'+[l cr] [i, tr]'} + 
( 10 ) 
in which any accented function is derived from the corresponding unaccented function 
by the substitution of the coordinates of the pole of Q* in place of the running co¬ 
ordinates. 
The following are, in the first instance, the integrals to which the method of this 
paper will be applied :— 
O) ( 
j/U'P^S 
(2) [ 
{(1-/P)MV/S 
(3) { 
(4) j 
P^P^S 
0) | 
j(p, a )(q, f3)(r, a+/3)r/S 
f 
(p, <*)[$, £] [a a+/3]c/S 
(6) j 
jp„P ? PAV7S 
Of these cases the first three require no special comment. The fourth case was for the 
first time solved by Professor J. C. Adams, in the Proceedings of the Boyal Society, 
vol. xxvii., pp. 63-71, although the result had been independently found and published 
by Mr. Ferrers in his ‘Treatise on Spherical Harmonics.’ 
Professor Adams, as well as Mr. Ferrers, made the discovery of the value of (4) 
by an inductive process, and Mr. Todhunter showed in a subsequent number of the 
Proceedings how the proof could be thrown into a compact form. The present 
writer was, however, convinced that the integral could be found deductively according 
to the method described above, and the simple character of the result led him to hope 
