SOLUTIONS OF DIFFERENTIAL EQUATIONS. 195 



The conditional equation is 



f /Nd^wl d f /Jj d^w\ , s 



\VZ^T) = tAv N-25"/- * (74) 



d_ 



dy {. V L dy 



Equations (72), (73) determine the nature of the solu- 

 tions N=o and L=o. 



This equation must be satisfied by a proper value of 



*/w, or otherwise equations (72) and (73) cannot be 



integrated. 



Example : — 



p z =i-y 2 (from Cayley). . . . (75) 



The condition of equal roots is y = + 1, which is a 

 solution of (75); but whether it is singular or particular 

 depends upon (72) and (73). In this case s + r=o, L=i, 

 5= Vi— y z . 



The equation (74) is satisfied by 



*/w = smx\/i — y % (76) 



Then (72) and (73) become 



dv 



-7- = —y sin x. 



dx u ' 



from which 



dv 



-rr- = COS X. 



dy 



v=y cosx, 

 therefore the complete primitive is 



c + y cos x + sin x V 1 — y z = o, 



and y= + 1 is a singular solution of the envelope species. 

 Example : — 



p 2 {i-x z ) = i-y 2 (from Cayley). . . (77) 



The condition of equal roots is in this case x— ± 1 and 



