ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 2o7 



where the constants a,, a.^, . . rt^, are determinate rational functions of 

 the other points, 



'•'■'., !/'\ (v = 1, 2 . . . ,7) ; a;V., y%.. (^- = 1, 2 . . . p) 

 find.'-,, »/.(/:= 1, 2 . . . q). 



The corresponding values of y are the remaining ordinates of the points 

 of intersection of the curves y {x,i/) = jO, and ti {x,y) = 0. 



(17) Abel, in the proof of his theorem, stated in art. 12 wrote : 



';'>('^') = [f{^',y)d^, 



■where/ {x,i/) is any rational algebraic function of .v and y. He then 

 considered the sum of such integrals 



i = M 



t ^ 1 



where .r- (i = 1, 2, . . . ^() are the points of intersection of the two curves 

 x(-*-")2/) = 0, 6{o:,y) = ; that is, the roots of the equation E(:<;) = 0, 

 which is obtained by eliminating y out of the two given equations. 



Of these points of intersection some may be stationary, while the others 

 are movable, the fixed points being independent of the parameters 

 10, V, tr, . . . (art. 13). Hence E (x) may be composed of two factors 

 Fu{x) and F(.«), of which Fu(.<:) does not depend upon u, v, iv, . . . 



Abel wrote the subject of integration in the form 



f\(^ ,y) 



A{^;y) x'{y) ' 



where /i(x,y) and f;{x,y) are integral functions of x and y and 



X'(2/) = ^^. 

 d y 



He found that V ^i.-'-) = 'y, /'('^■' /■) c^.y. = v, 



tZi <=i A{^i,yi)x{yi) 



where v may be a constant phis an algebraic function plus a logarithmic 

 function. (A concise expression for v, due to Rowe, is given in art. 20.) 



(18) After restricting tlie functions /\{x,y) , f.2{x,y) and Fo(.>j) in such 

 a way that the logarithuaic and algebraic functions of the expression above 

 disappeared, Abel found that the function f\{x,y) contained a certain 

 number of arbitrary constants, a number which depended only upon the 

 nature of the curve x(-'''I/) = ^- '^^^^ number he designated by 

 y(=^>, of art. 13). 



In the equation 0{x,y) = q^, + q^y + q.pf- + . . . + q,^_^y"-^ = 0, a 

 certain number of the coefficients of x in the functions q are supposed 

 indeterminate. Denote these by o, (/,, «2> • • • We saw above that the 

 upper limits .i;,(i =1, 2, . . . yu) of the integrals in (I.) are the roots of 

 the equation E(.r) = 0, and mfiy be expressed as functions of the inde- 

 pendent quantities «, «,, a.^, . . . , of which there are, say, a. 



Let these functions be : 



»i =/i(«> «ij «2. • • • )' '^'2 —f-i^h «:, «o, ...).. . .T^ =/^(«, (ix-, «2, . . . ). 

 1897. s 



