SOLUTIONS OF DIFFERENTIAL EQUATIONS. 198 



Hence w = o is a singular solution o£ (63) of the envelope 

 species. 



16. By reference to equation (11) and by dropping the 

 affixes to the variables (v) and (w) it is easy to obtain 



C (dvY fdw\ z ~) 2 f dv dv dw dw \ 



\ \dyj \dy) j^ r\Jt dx dy dx dy S 



+ ^(£)*-(£T= a • • • <m 



This equation, which is of the second degree in (p), is 

 expressed in terms of its primitive c + v + \/w = o, and 

 always has a singular solution w = o when (v) is not a 

 constant. 



A differential equation of the second degree and the 

 first order, which is not of the form (64), cannot have a 

 singular solution. 



Professor Cayley says (see ' Messenger of Mathematics/ 

 vol. vi. p. 23) that 



L^ + 2Mp + N = o (65) 



has not in general any singular solution when L, M, N 

 are rational and integral functions of x, y. 



If I apprehend correctly the meaning of this statement, 

 it seems difficult to reconcile it with equation (64), the 

 terms of which can always be made rational and integral 

 functions of a?, y by proper assumptions of (v) and (10). 



Now, equation (65) admits of the form 



Ly + 2MLp + M z -N=o. . . . (66) 



The condition of equal roots is N = o, and this equation 

 is a solution of (66) if 



, , dN T dN 



M W =L ^ ( 6 7) 



