APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 99 



Gergonne, The solution of the general problem was given by Pon- 

 CELET in his "Traite des proprietes projectives," p. 352, 2d ed.. 

 p. 340. For these references see Fiedler-Salmon's Analyt. Geome- 

 trie der Kegelschnitte, ii, p. xii, (88). 



II. PROBLEMS OF CLOSURE ON THE TWISTED QUARTIC OF THE 

 FIRST SPECIES. PROJECTIVE THEORY. 



§6. Generalization of Abel's Theorem.' 



1. In order to extend Abel's theorem to twisted curves, assume 

 a plane curve C of deficiency p with the equation 



/(«',2/)-0, (1) 



and a point with the coordinates 



X=(^(a., y), Y=a/r(^, 2/), Z=x(^, V\ (2) 



where <^, i^, x ^"^^ rational functions of x and y. If x and y vary 

 according to (1), the point (2) describes a twisted curve V which 

 corresponds point for point to the curve C, provided </>, a/t, ^ have 

 not been chosen in some particular manner, which is supposed to be 

 excluded. Hence, x and y are, inversely, rational functions of X, Y, 

 Z. Every integral of the form 



'a^X+yScZY+7^Z (3) 



J' 



taken along F, where a, /3, 7 are rational functions of X, Y, Z, by 

 substitution (2), is transformed into an Abelian integral 



J^ 



B.{x^ y)dx 

 with respect to the curve C. On the other hand let 



n(X,Y,Z):::=£i^l^^ (4) 



"^ ' ' ^ Q(X, Y,Z) ^ ^ 



be a rational function, P and Q being of the same degree. Replacing 

 X, Y, Z by <^, i/r, X-, we have 



(») See Appkll et Coubsat, loc. cit., p. 432-434. 



