DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 113 



The coefficient of sin m\ in V is :S(V;;7(;;' + V™,/t'l'„), 



X„ S(Xr/C + X-„A!.'„), 

 Y „ 2(Y;r<y;;'+Y-„i/™„), 

 z„ S(z;;vc+Z"_vr„), 



in which n takes all integral values for a given value of m. 



In a portion of his work, in which he treats of the definite integral of 

 the product of two Legendre's coefficients, Professor Adams proves the 

 well known formulfp, which I believe were first proved by Legendre, that 

 when n and n^ are different from one another 



and that when ;? ^= n, 



He also proves that if 



d'"l? 



Hence if n and ?ti are not equal 



J'_ q;:' Qi dfi = 0. 



But if n^-=.n, then 



Hence if 

 it follows that 

 and, when ?i=?i,, we have 



It is also shown that 



H" - {n-m)\ Q,, 



"~1.3.5...(2«-1)^" 



And therefore, when n and «, are not equal, we have 



1898. 



