With respect to the development of 



(1 2r(cos cos 6'+ sin sin 0' cos ^>) + r 2 )-*, 



it is shown that the coefficient of r n cos i<$> may be put in either of the 

 two forms, 



1^2 (n i) I 2 (w + i) ^ ' v 31 ""./ 



or 



. .^:'I^ ) ':.^:'!l+o< si " 8 >'> i "' ) ')' e - i9 ' -'(*-- ^r- 



where represents the operation sin sin 0, and the factor 2 is in 



each case to be omitted when i=0. (This coefficient is a solution of 

 the equation 



of which the complete integral may be expressed in the form 

 (sin 0)- M (sin ^ sin 0)" "'(sin fl)*^ + C 2 p0(sin 0)- 2i _,), 



at least in the case in which i is an integer not greater than n, for 

 which case this form is here demonstrated.) 



If it be assumed that the solution of (2), obtained on the suppo- 

 sition that n is an integer, may be extended to the case in which n is 

 a general symbol, it follows that the solution of (1) will be obtained 



from it by changing n into r . This would give 



which is easily shown to be equivalent to 



where p=r(sin 0)" 1 , but p is to be treated as a constant till after all 

 operations. 



