100 UNIVERSITY OF COLORADO STtlDIES 



U{X,Y,Z)=U,{x,y), (5) 



IIj being a rational function of the point (a?, y) whose zeros correspond 

 to the points of intersection of the curve T with the surface S 

 having the equation P(X, Y, Z)==0, and the poles to the points of 

 intersection with the surface S' represented by Q(X, Y, Z)=0. 

 Hence, by Abel's theorem, it is possible to express by algebraic and 

 logarithmic quantities the difference between the sum of the values 

 of integral (3) at the points of intersection of the curve T and the 

 surface S, and the sum at the points of intersection of T with S ' . If 



(3) is an integral of the first kind, then also j R(a', y)dx is of the 



same kind, so that in this case the theorem of Abel may be stated as 

 follows : — 



The sum of the values of the Ahelian integral of the first kind 



CadX-\-/3dY-\-ydZ, 



attached to an algebraic twisted curve F, taken from a fixed origin 

 to the points of intersection of this curve with a variable algebraic 

 surface remains constant, if the coeficients of the equation of this 

 surface vary in an arbitrary manner. 



2. Suppose now that the curve T is also represented by the 

 intersection of the algebraic surfaces 



f{X,Y,Z)=0, 



/,(X,Y,Z)=0, 



(6) 



of degree m. Then 





C!") 



From thig 



^Y = 



Bx Bz ax sz 



BY BZ BY BZ 



(') As before B stands here for the partial differentiation sigm. 



