J^ 



84 UNIVERSITY OF COLORADO STUDIES 



than logarithms; hence the sum S is a rational function of the 

 coejjflcients «„ a^^ . . . a^, increased by a sum of logarithms of rational 

 functions of the same coefficients: 



S=p-\-'2Alog<T, (9) 



the A's being constants. 



2. We shall specialise this theorem for the case of Abelian 

 integrals of the lirst kind. Such an integral 



(f . y) 



R [x, y) dx 



K, 2/o) 



remains finite for every point (a?, y) of the corresponding Rieraann 

 surface. The function (9), which represents S, must remain finite 

 for all finite or infinite values of the parameters a. But a function 



/o (a) + 2 A log a (a), 



where p (a) and <r (a) are rational functions of a, which remains finite 

 for all values of a, reduces necessarily to a constant. 



Hence, in case of an Abelian integral of the first kind, the sum 



2 fR{x,y)dx, (10) 



where («„, y„) designates the points of intersection of the curves [1) 

 and (5), which vary with the d's, is independent of these parameters 

 and (^excepting sums of multiples of certain fixed periods which 

 may always be introduced) has a constant value. 



3. The Abelian integrals of the first kind with respect to a curve 

 of the mth order are, as is well known, of the form 



'Q {x. y) dx, (11) 



r 



r 

 h 



6 is here and subsequently used as a partial differential sign. 



