[SULLIVAN] CONCERNING THE INTEGRALS OF LELIEUVRE 179 



This is a linear equation of the first order; its integral is 



^ *" ^ (P+qyv(P)dp-h2Uq)\ 



(p+qy 



2(773+^3) 2 



^2+^71, 



(P+qy (P+q) 

 where 



v{p) = m'{p), m{p) = m'(P), m{p) = vs'{p)- 

 But 



dq (p+qY 

 and therefore 



2 



(p-\-q) 

 or (on changing the notation sHghtly) 



The function f(g) is not however entirely arbitrary. We have seen 

 by (21) that 6 must satisfy the equation 



\X do/ 



dqW dq> 

 which implies the following condition on ^(q) : 



From this we conclude that 



r=o. 



and therefore 



^ = aq"-\-bq-\-r. 

 The function now takes the form 



• d=r,'+2{ap-b)--^L-\-(ap^-2bp)-\-c] 

 p-\-ql ' 



dp p-\-q 

 where 



(T(p) = v{p)-\'p{ap-2b) + c. 

 We can verify at once that 



0= a — ff 



p-^q 



satisfies the condition 



r" = o. 



Sec. Ill, Sig. 14 



