306] 



TWO TYPES OF SOLUTION. 



561 



71 V dx dz 



(13), 



where 



P n = 



(Ar) + (w - 1) ^ (Ar)} ......... (14). 



To obtain the corresponding formulae for the solutions of the 

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



- 

 dr 



dx 



dx 



1 J dx 



dx 



, 



1 



The remaining terms give 



(r ^ - l) u = (n + 1) {Ar^ (Ar) + (n - 2) ^ M _ 1 (Ar)} ^ 



-_ 



and 



...... (17). 



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

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



r P&amp;gt; , = A n |&amp;gt; + B n r^ ^ ^ + C n ^ 



d &amp;lt;l&amp;gt; n 

 dy r *n+i&amp;gt; 



rp rz =A n 

 where 



..(18), 



_ 2(71-1) r 2 

 w ~ A 2 271 + 1 



C n = JJL(U + I) {hr^ n^ (hr) + 2 (n - 1) ^v-i 

 D n = - 



L. 



...(19). 



36 



