Ill 



and therefore, differentiating (X') as if «, were constant, 



c?" /dO,\'> 

 d»,"\da,) 



d" fd0,\n+ 1 



rf» (fy, /rf(p, \" + 1 



rf< 



5," " £?«; \,c/«, / 



■*■ ("X* r« -4- I "i» ■)- Wz a. /t>TT- ' ^ 



that is by (fl'). 



[» H- rj" -f- 1 (■*/ + ^)"+ ' ' 



1 dV 

 It. dz, I ' I 



f 



I 



Ca^e ©/"a System of Revolution. 



6. Another particular case of the partial differential equation 

 (A^, deserves to be considered specially ; namely the case of sys- 

 tems of revolution, symmetric about some single ray. In this case, 

 if we take for the axis of z, the ray which is the axis of the system, 

 V will be a function of z and of x* + ?/' ; and if we put 



a^ + y^=n, (O ) 



we may in general suppose V developed according to positive integer 

 powers of ri, in a series satisfying the condition, 



To integrate this partial differential equation (P'), which is a 

 particular case of C-4), we may employ the corresponding case of 

 the general system of equations (D), (E), putting for abridgment 



»' + /i' = i. (Q') 



