1882.] 



On Abel's Theorem and Abelian Functions. 



289 



or, what is the same thing, by the co-ordinates x of the points of 

 intersection of the three surfaces represented by ¥ m , F n , Fp. 



Some preliminary considerations (in connexion with §§92 sqq. of 

 Salmon's Higher Algebra) are adduced in reference to the eliminants 

 of the three equations in each of the variables ; thus if X be the equa- 

 tion in x obtained by eliminating z and y, it is expressed in the form 



X= BjmFot -f B^F^ + B P F 



Pi 



which afterwards proves useful. Then the ordinary case (above 

 referred to) of Abel's theorem is treated on the lines laid down in 

 Clebsch and Gor dan's treatise on the Abelian functions ; and under 

 the guidance of this the more general form is investigated with the 

 result 



s f ™ =e rJ-1 • 2 ( E log F, ) + 0, 



V y, z J 



being the symbol introduced by Boole. 



The remainder of the section is occupied with the discussion of two 

 examples of this theorem. In Example I, by the assumption of suit- 

 able forms for F /tt , F„, F p , it is proved that 



E«) + E«) + E«) -E(tt, +u*+Uo)= - SJfABG 



v i/T v 2 y-r v v 2 ^ sJ (A2+B2-A;2C 3 ) 2 +4fc 3 A2G3 



where E is the second elliptic integral and A, B, C are given by 



As l +Bc 1 + CcZ 1 =l, 

 As 3 + Bc 2 + Cd 2 =l, 

 As 5 + Bc s + Cd 3 =l, 



and s, c, d stand for sn u, cn u, dn u respectively. The corresponding 

 expression for the third elliptic integral is stated. 

 In Example II an expression is obtained for 



E(?< 1 + w 2 + .... +w 7 ). 



In Section II the addition theorem for the functions presented in 

 Weierstrass's memoir in Crelle, t. lii (1856), p. 285, is investigated. 

 It may be pointed out that the fundamental equations in the theory 

 occur as natural examples of the more general form of Abel's theorem 

 proved in Section I; but the equations so obtained are identical with 

 those used by Weierstrass, and this case, therefore, does not belong 

 distinctively to the form of Abel's theorem connected with the curve 

 of double curvature. On this account the simpler form is used on 

 the two occasions (in §§ 14, 19) when required. 



The theory is worked out at considerable length, and the necessary 



