306] 



TWO TYPES OF SOLUTION. 



561 



where P n = /* {hr$ n ' (hr) + (n - I) ^ n _, (hr)} ......... (14). 



To obtain the corresponding formulae for the solutions of the 

 Second Class, we remark first that the terms in p n give 



d p n (U} 



2n+l dx Zn + ldxr^" 



The remaining terms give 



(r - l) ' = ( + 1) [^' M (hr) + (n - 2) ^ M (fcr)} ^ 



- n (M/ n+1 (Ar) + n+ n+1 (hr)} h*~ ~ ^ ...... (16), 



and 



_ ( XU > + ^ + ^) = n (n + 1) ( 2n + 1) ^ n (Ar) n 



...... (17). 



Various reductions have here been effected by means of Art. 305 

 (7), (8), (15). Hence, and by symmetry, we obtain 



where 



n 2n+l' 



(hr) tyn+i (hr)} 



36 



