290 On AbeVs Theorem and Abelian Functions. [Nov. 23 r 



formulas are obtained in a manner somewhat different from that of 

 Weierstrass. 



The fundamental equations being 



?/ 2 — P(x)=yt— — d])(x— a 2 ) a p ) = 0, 



z 2 — Q(V)=z 2 — {x — a p+1 )(x— a p+2 ) . . O— a 3p+1 )=0, 

 0=M?/ + N;z 



Avhere M=xp + M 1 xp~ 1 + .... f M p _^-f M p , 

 N= N^p-^ +N p _ 1 aj+N'p, 



the equation giving the roots x is 



MV-N¥=0. 



The Sp roots are denoted by ,t 1? x 2 , . . . , x p ; g^, £. 2 , • . . , £ p ; Pi? j?2> - • r 

 j; p ; and there are obviously p relations between them. Writing 



B(*)=P(*)Qto. 



^ M " =i f=xJ«x v'Eg 0<=1 ' 2 ' • • • ' P) ' 



and v, w corresponding functions of gf,j9 3 it is shown that 



Writing, with Weierstrass, 



0(»)=(»— ^2) • • • • (*— »p), 



— Q(«r)=^ 0—1, 2, />),■ 



p(a p+ ,)=z P+5 («=i,2,...r,/.+i), 



then 2/) + l of the functions of the theory are given by 



Z r aZ r 2 =0(<x r ) 

 for values 1, 2, , 2/> + l of r. Then if 



it is proved that 



If V, W are respectively the same functions of the £'s and p's as TJ 

 is of the as's, then the theorem 



U+V+W =M I%A— al m (u)al m (v)al m (u+v) 

 is obtained in § 21, a verification being furnished by expansion in 



