[SULLIVAN] CONCERNING THE INTEGRALS OF LELIEUVRE 183 



On eliminating ai, ao, as between these equations, there results the 

 relation 



(49) {p-\-qy W<J2"a^"')-2{p+q) {<T,(J2"cTz"')^-2{a,a','c^"') 



m -''^•''' 



From equations (9) and (26), it can be readily shown that 



(50) 



-{p-\-qyVMVA = {p + qyWa2"<Tz'")-2{p + q){a,a2"(r3"') + 2iai<T2'(j/'). 

 Hence 



(51) g (p, q) + (p+qy VmVA = 0. 

 From these equations we derive at once the result 



(52) fg^Q 



dq^ ' 



and therefore 



g{p,q)=k{p)q'--2l{p)q + m{p), 

 a quadratic function in q. 



We have now before us the equations necessary for the discussion 

 of the various configurations that may occur. 



First. 



If çi, q2, qs be three arbitrary values of q, then from (47), we have 



(53) ai(gi)(ri + a2(çi)o-2+a3(gi)(r3=7ri(^), 

 ai (go) <ri + £12 (§2) <T2 + as (g2) 0-3 — 7r2 (p) , 

 ai iqs) (Ti + 02 (qs) <T2 + as (qs) «rs = 7r3 (/>) , 



where wiip) is a quadratic function. If then (ai(gi) 02(^2) «3(53)) is un- 

 equal to zero, (Tiip), (T2(p), (Tsip) are functions of the second degree, and 

 A vanishes identically as well as T". The integral surface (S) is there- 

 fore a ruled quadric. 



Second. 



If the determinant (ai(gi) a2(g2) 03(^3)) vanishes, we have yet an 

 exceptional case to consider, namely, when g(p, q) vanishes identically. 

 In this case each of the determinants {ai 02" as") , {(Ti<T2'<t/'), {<xi(X2" (tz") 

 must be equal to zero. The functions ci, 0-2, 0-3 must therefore be con- 

 nected by a relation of the form 



aCTi -f- Ô0-2 + C(73 = 0. 



But this relation implies a linear relation between Bi, 62, 63 which was 

 excluded at the outset. 



Third. 



The function g(p, q) does not vanish identically; but it vanishes 

 for 



(54) q = l{p) ± VP{p)-k(p) m{p). 



