:MUi;riPLicA'riox of elliptic functions. 153 



liiuLself, und tlieii applied to tlie iiivestigiiticju of tlie multiplication- 

 foi'inuJa'. The ])i-esent paper i« thus composed of two parts, each of 

 them having referenœ to the tlieory of mnltiphcation hut otherwise 

 unconnected. 



Throughout fhe paper, \ have adopted the notation of Fund. 

 Nora, the only exception l)eing tViat I write ^1, fi.,, 0.^, H instead of ft 

 and II, wherehv 1 follow dacobi in his lectures.* 



Part First. 



Denote the fundamental elliptic irrationality by s and the corres- 

 ponding Riemann's surface by T. Let S stand for an algeljraic func- 

 tion of i, one- valued on tlie surface 7', and of the order </, having for 

 its zero- points 



Of, {.--'m I -s^}, fi = \,'2, 3, q, 



and for its intinity-points 



-'■ IC" i '^"f» î^ = 1, '2, 8, (J, 



then iVbcl's theoi'eni is expi'essed by 



(1.) y ^^ - y ^ = 0. 



Xow .S is necessarily of tlie Inrni 



i" + ^ ■s- 

 U 



where l\ (J, 11 are integral functions of ï of the degree <[, <[-- and ([ 



* Jacobi, Gesaumielte Werke, Bd. I, p. 501. 



