[SULLIVAN] A CERTAIN PROJECTIVE CONFIGURATION 153 



Further, it may be shewn that a system of equations of the form 

 (1) whose coefficients are analytic functions of u and v has precisely 

 four linearly independent analytic integrals 



3;W=/W(«, z;) (yfe = l, 2, 3, 4); 



provided, of course, that the integrability conditions (2) are satisfied. 



Any set of four solutions of (1) for which the determinant 



[y Ju Jv Juv I is different from zero is called a fundamental set of 



solutions; and the most general solution may be expressed in the form 



where the c's are constants. Thus any other fundamental set of 

 solutions, sayy(^n^ = l. 2, 3, 4), will be of the form 



^(k)=cu,yi) + Ck2y2)+Ck3>'(3) + Ck4y*), (kki|4=0), (^ = 1, 2, 3, 4), 

 where the c's are constants. 



If, then, 3;^^), ;y^^\ 3^^^^ 3/^^) be interpreted as the homogeneous 

 coordinates of a point in three dimensional space, the locus of the 

 point Py is a surface 5; and, furthermore, this surface has the para- 

 metric curves y = const, and « = const, as asymptotic curves. We 

 shall denote these parametric curves by V and T" respectively. 



Conversely, if an arbitrary non-developable curved surface 5 be 

 given by the equations 



yfe)=/W(^^, y) (;^ = 1, 2, 3, 4) . 



(the asymptotic curves being parametric), then a system of equations 

 of the form (1) can be determined which has S as its integral surface. 



It follows from (3) that the most general integral surface of (1) 

 is a projective transformation of the surface S, and that the coefficients 

 of (1) have the same significance for all such surfaces. Now the form 

 of the coefficients of (1) depends upon the particular analytic repre- 

 sentation used; since, if we apply to (1) the transformation 



(4) u = a{u), v = ^{v), y = \(u,v) y, 



where a, j3, X, are arbitrary functions of their arguments, the surface S 

 and the parametric curves (F', T") remain unchanged while the 

 coefficients of (1) assume the new values (a', b,. . . etc.,) where 



(5) â'=^ a\^^-^b, .... etc., 



Those functions of the coefficients (a, a',..., c, c') and their 

 derivatives which remain unchanged by a transformation of the form 

 (4) are called Invariants; and those functions of (a, a',...c, c' ; y) 

 and their derivatives unchanged by this transformation are called 



