ON ELLIPTIC AND HYPERELUPTIC FUNCTIONS. 311 



From the formula 



0(0, k) ~ Vy^ ' d{0,i:') ' 



given by Jacobi in the ' Eundamenta Nova,' p. 165, Eoseuhain deduces the 

 following (p. 395) :— 





V- 



where 



He theu enunciates the following theorem : — 



where 



, , log,i)log.,2--t--^' log^^'log^5'— 4A'' 



log U lOg^ « = Tf- = , • T i » 



^^■^ ^^-^ log^g' log^2 



, log^plog^5-4A' ^ log.p' log, 2'-4A^^ 

 log,, = ^^ , log, 2= ^^. . 



iVA . , ttA' 

 A'=i-— :> iA= 



log^/ ^ logeP" 



^y , wlog jj — 2Ai' 



V = r ■, w = , 



log.JP logj^ 



■kv' iu' log, «' — 2iA.'v' 



v=i „ "■' — 



log,i>" log,j?' 



To prove this theorem, which is enunciated without demonstration, I ob- 

 serve that 



v + 2nA= 1 (v' — 2mA ), 



^ log, « ^ ■' 



^Oe 



P 



according to supposition. 



Wherefore by formula (1) of this section 



log p log q-iA.^ 



00 - 



log,? A / - i ^^'(''' + -"'^ '^ ) 



V log,^ 



(by th(- t'i>nnula just derived from tlie ' Fundamonta Nova '). 



