[SULLIVAN] A CERTAIN PROJECTIVE CONFIGURATION 157 



But this relation can be reduced to a much simpler form by means 

 of a transformation of the type (8). In short, if we apply the trans- 

 formation 



u = a(ii) = \ J U du, V = fi{v) = J V dv, y — c J au ^v y 



to system (6) and denote the coefficients of the transformed system by 

 {a, b, etc.,) we shall find 



(11) -^ = ^ ^ = 1 



The most general transformation of ihe type (8) that leaves this 

 relation undisturbed is evidently 



u = u, v=v, y = cy, 



where c is an arbitrary constant. 



Let us assume that the relation (11) has already been effected; so 

 that the conditions 



become 



a' =b = (j){u, v), 



where ^(«, v) is a solution of the equation 



(12) 



du dv 



Now this equation can readily be transformed into one of the Liouville 

 type, from which the general solution of (12) is found to be* 



where U and V are functions of u alone and v alone respectively, and 

 where f/4=0, F' + O. 



On taking account of equations (9), (11) and (12), the conditions 



lead to the following expressions for/ and g in terms of the function <^: 



(13) 



dn- V dît J 



1 â- log </) _ / Ô log V<^ ^2 



^ dv- 



-<^u-h-~r^-{p^^^^ 



We must now determine whether or not the integrability condi- 

 tions (7) impose any further restrictions on the function (/>. The 

 first two integrability conditions become (in virtue of (11)) 



*S, p. 176. 



