350 REPORT— 1869. 



we shall obtain from (11) 



u,(q,y — a, — l)w(q,y — fi—\) 



From this Heine deduces easily 

 



fl+ ''^^'' I— ^'^ "> q- qA= <o(q-,l)t„(q-,0) 



V log, 77 \ loge?/ 



■whence wc have 



0(,r) = 0\(O)* . l-'2q cos 2.v + (l 



2^q 1-r 



J. (l-2q COS 2.v+r/) ^2 , (1-% cos 2.f+g")(l— 2(/^ cos 2.i- + /) , "1 

 t (l-'/)(l-5') ^ (l-iO(l-'/*)(l-20(l-V) ^ J' 



and a similar formula for 6^.v. 



Heine also gives some formulae for the multiplication of elliptic functions, 

 of which I shall give one here. 



From the equation 



where 



C: 



(2», a)w{q", a- - j (o fq", «_ =Y . . o,fq'^, (i- ''^—l\=coj(q, no), 

 {(l-q"Xl-q'-")(l-q'")...}" 



(l-2)(l-rXl-5^)--. ' 

 • e have, where (n) is odd, 



n e 1 q",-±!^(b-mtiosq) \ = ^ y he{ q, ), 



and when (n) is even, 



m=n-i r 9K , ., ."1 «■(— l)^"e'''" ,„ / 2K^\ 



n e r/',rL»(6-mlog,7) =_ J--' m(^,=i^\ 



'where 6(7",'^^—^) hears the same relation to g" that e|^l-^ ) does tog, and 



j^_ \a-q^-»){l-q^-)...\ 



Section 8. — The comparative simplicity of the functions naturally sug- 

 gested to mathematicians the utility of adopting these series as ground-forms 

 in the theory of elliptic transcendents. These functions have therefore been 

 in these last times the subject of many investigations. Among the memoirs 

 relating to these series, thi-ee may be particidarly mentioned as having re- 

 markably contributed to the advancement of our knowledge of their proper- 

 ties. These three papers are Jacobi's memoir " Sur la Eotation d'un Corps," 

 in the thirty-ninth volume of Crelle's Journal, a paper by Krusemarck, " Zur 

 Theorie der elliy)tischen Functioncn," in the forty-sixth volume, and a paper 

 by Richclot, " TJeber cine Merkwurdige Formel in dcr Theorie dcr elliptischen 



* For the value of 0i'(O) see a little further on. 



