[SULLIVAN] CONCERNING THE INTEGRALS OF LELIEUVRE 177 



It therefore follows that 

 (18) da d%. db d%^ 



dp dq^ dp dq- 

 Now a and b can not be constants, since di, d^, 63 are independent; they 

 can not be functions of q alone, since A does not vanish. If A and T" 

 both vanished identically, the surface would be a ruled quadric. 



On eliminating — and — between (16) and (18), we find 

 dp dp 



(19) 



<dq 





0. 



dq 



The integration of this equation leads to the relation 

 (20) a{p)9r + 0(p)d2=l, 



and this is the restriction imposed on (A) by the conditions of the 

 problem. 



If we now differentiate (20) with respect to q and the resulting 

 equation with respect to p, cognizance being taken of (A), we find the 

 relation 



X dp dq 



Thus 



(22) l^^_^^/l^ = o 



X âg2 Q^ Qq \^J ~~ 

 since a-Çp) can not be a constant. 



Now derive (22) with respect to p, then (on taking account of (15)) 



dq^ dp Vx/ X dq dq \\/ dq dpdq \\/ 



or what is the same thing 



(23) ^A/^A-L^d- 



dq^ dp \X/ dq \ dpdq / 



+ — \1+ — /=0. 



