ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 267 



After some reductions he was able to express the n — 1 integrals of the 

 system (S'.) in the form, 





F(a;i) 'a.-a;!^ F'(a;2) '0,-0:2'^ • • • -r p.^^j „^ _ ^_ 



■= A.2„_l "{(ill), 



(A = 1, 2, ... n - 1). 



One recognises at once much similarity between the quantities employed 

 here and those later used by Weiorstrass in Crelle, bd. xlviii. 



(26) Jacobi (' Demonstratio nova theorematis Abeliani,' Crelle, bd. xxiv. 

 p. 28), derived the n — I algebraic integrals in a different manner than 

 that given above, and at the same time he established a new proof of Abel's 

 theorem. Instead of the system (2.) he introduced n differential equations 

 with n variables X^, Xn, . . . X„ and the variable t : 



(2.") 



TyW)"^ vTPT) + • • • + v7(Aj =^' 



(i = 0, 1, . . . Ji - 2) 

 with the equation 



^^CM^J +^^C!^+ + ^""'^^" = dt ■ 



wherey(\) is an integral function of the 2w — 1th degree in \. 



For brevity let N. = (A. - X,) (X, - X^) ... (X. - X„), 

 where the vanishing factor X, — X, is omitted ; 



and 1/ = v^(w2. — X,) (m — X^) ... (to — X„), 

 where to — X is any factor of the function /{X). 



The following lemma is next proved : If if/ (X) is a rational integral 

 function of the 2n—2th degree in A, then 





Making use of this lemma, Jacobi found an algebraic integral of the 

 system {^".) in the form 



y ( n//(^ i) + n//(X 2) + . . . + ^fJK ) ) = Const. 

 \(m-Xi)Ni (to-A2)N2 (TO-A„)]Sr„/ 



Corresponding to the 2u — 1 factors vi — X of the function / (X), there 

 are 2n — 1 integrals of the form just written, of which n — 1 are sufficient 

 to give the algebraic relations between the n variables Xj, X,, . . . X„. 



Haedenkamp (Crelle, bd. xxv. p. 178), specialised the general case, and 

 found by geometric considerations the two complete algebraic integrals of 

 the system (a.) above. 



(27) In each of the n — 1 integrals derived by Richelot appear two 

 roots of the equation/ (.r) = 0, while in Jacobi's solutions there is found 

 only one ; and if imaginary roots enter/(.T), the integrals found in both of 

 the methods just given have imaginary forms. 



Richelot (Crelle, bd. xxv. p. 97) found another method of solution 

 which also depends upon t^^o roots of the equation /(a;) = 0, but has the 

 property of remaining real when for these two roots any two conjugate 

 roots are substituted. He succeeded further in finding a system of n—1 



