ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 255 

 Let R {Xyy) be any rational function of x and y, and form the sum 



2^ 'R(x„y,)dx,. 

 -KJ xj>,y° 



For the present discussion it is in every respect sufficient to consider 

 only one parameter ?t, and to specify the function 8(x,y)=:0, which we do 

 by writing 



e{x,y)=(l>(x,y)-u\P{x,y)=0, . . (1) 



or M= - \ '^^ 



.0(^^ 



4'{x,y) 



The rational function -^^— Stakes the value Uq as often as it does any 



xP{x,y) 



other value u. 

 Writing 



(I.) to.= 'R(x„y,)dx., 



we have 



«-" '=■'' x,,y, 

 (II.) 2«'.= 2 R(«4/.)^^- 



In the expression (II.) we shall study w as a function of u. 

 From (I.) 



dw^ dw, dx,_^, dx, 



du dx^dk^^'^-'^'^d^ .... (2) 



Differentiate (1) with regard to u, and we have 



WoxJx=x,\^y )x=x,dx.\du 



^ y=y. y=y. 



y=y, y=y. 



Sincef ^ ) -r-'+i^] =0, we have, after substitutine the 



\oyJx=x,ax, \oxJx=x, ° 



y=y. y=y, 



value of -^' from this equation in (3), a formula for — which is rational 

 ax. du 



in a;,, y.and u. We may therefore write ~=S(x„7/„2c), where S denotes 



du 

 3. rational function. 



dw 

 du 



Further, -^'='R{x.,y.) S{x„y„u)=T(x„y„%i) where T is also a rational 



function. 



-^^"^^^^'Sm^ ^~S^(^"y"^*)=^(")' ^^y- ^^^^^ *■(") '8 * rational 

 «=i «=i 



function in u. 



