305] PRELIMINARY ANALYSIS. 557 



we find, in the solutions of the First Class, 



(11); 



these make 



?) = -n(n + l)^n(hr)x n (12). 



In the solutions of the Second Class, we have 



d 



= - <2n + 1) A^. (Ar) (y ~ - 



1 ' = -(2n+l)h^n(hr)(z~-x^ r }<t> n \ (13), 



and therefore 



^^O ..................... (14). 



In the derivation of these results use has been made of (7), 

 and of the easily verified formula 



To shew that the aggregate of the solutions of the types (3) 

 and (6), with all integral values of n, and all possible forms of the 

 harmonics <f> n , Xn, constitutes the complete solution of the proposed 

 system of equations (1) and (2), we remark in the first place that 

 the equations in question imply 



(V 2 + A 2 )(#w' + 2/?/ + ^/) = () ............... (16), 



and (V + ^)(f + yV + O = ............... (17). 



It is evident from Arts. 266, 267 that the complete solution of 

 these, subject to the condition of finiteness at the origin, is 

 contained in the equations (9) and (12), above, if these be 

 generalized by prefixing the sign 2 of summation with respect to 

 n. Now when xu + yv' + zw' and x% ' + yj] + z% are given through- 



