p-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 815 



or 



(jzf) V V-*X w -i«)£=A (36), 



where 



u = =p J { & - ib - 4ay[x 2 } . 



Since ad^O, it is obvious that the necessary and sufficient condition that (36) be 

 algebraic is that /3 be a perfect square, that is, that (a + 2cf- 16b be a perfect square. 

 It is assumed, of course, that a, b, c are commensurable. 



The condition that (29) may have a singular solution is 



c2+ac-4&=0 (37). 



This is not in general satisfied when the primitive is algebraic ; although the primitive 

 is always algebraic when (37) is satisfied; viz., in this case, (a + 2c)' 2 - 16b reduces to 

 (a + 2c) 2 — 4c 2 — 4ac = a 2 . 



As this contradicts a well-known result of Cayley's, it may be well to examine a 

 simple particular case of a differential equation which has an algebraic primitive, but 

 has no singular solution. By means of the above results we can construct an infinity of 

 such cases ; the fact being that it is the exception and not the rule that there is a 

 singular solution when the primitive is algebraic. 



Example of a Differential Equation which has an Algebraic Primitive but has no 



Singular Solution. 



Consider 



fy +lx*-$xp+p* = (1), 



the jp-discriminant of which is 



y=0 ..... (2), 



and gives no solution of (1). 



We may write (1) in the form 



P = lx±J(-Sy), 

 or, if x 2 = £, 



Put y = v£, and we get 



which may be written 



Let 





WHO- 



^+2v = -h±J(-Sv), 



Mv _ L f ^ = . . . . (3). 



u=Tj(-12v), 

 u 2 = -12v; 



