SOLUTIONS OF DIFFERENTIAL EQUATIONS. 183 



Equations (21) and (22) become by substitution 



C dv dz, ~\ dw 



dy \dx dx b J dx __ ( . 



cfo (dv dz\ ~) dw~ ' 



c + v + z\ogw=o (32) 



Now (31) is satisfied by w = o, which, however, is not a 



singular, but a particular solution by taking (c) equal 



to infinity. 



From (32) 



, c-f v 



LU£ 



\ W ~ 





3 



z 





•"• 



w = 



= € 



c+v 



z e 



~" C V 



£Z £Z 





,*. 



W- 



= 0, 



when c = 



infinity. 



Let 



f(w) — w n , where (n) is an integer. 



I 



niv n 



Substitute these values in (21) and (22), then 



/dv -dz\ 1 -- dw 

 7 n[- r + w n ^-)w nj rZ-r- 

 dy \dx ax/ ax __ . . 



dx + ~lTv l -dz\ 1-- 7Ho" 0> ' ' t33j 

 n\- r + w n - r )w n + z~r- 

 \dy ay J dy 



1 

 c + v \-zw^—o (34) 



This equation, viz. (33), is satisfied by w = o, which is a 

 singular solution, if it is not contained in the complete 

 primitive (34) by giving a constant value to (c). 



