356 



0,0 0,0-=- = 



fx sin X 



KEPOKT — 18G9. 



-4smxS, ^^ ' 



'i l—2(/-*' cos 2^7 + 2'''" 



00 0,0 -^ = tan A- +4 sin 2.^2 



(_l)s^2s 



— 

 gx 



1 1+222* cos 2.r+2''*" 



0,0 qJ^ = -1_ + 4 cos .rS" ^'(1 + ^^ 



cos a? 



"'i l + 222ieos2a' + 24** 



Oo 6 — ^ -L 4 cos S V <.i V I i / _ 



^ gx cos.i' ' l + 22'2* cos 2a' +(/■«» 



^"^^^■" <'l+222»-icos2.r + g^»-2' 



0..O 03O-|:^=4cos< ^_^,C(^ + t':\. . . 

 - 2 ;i;v " 1 + 22"*-' cos 2^7+2^*- - 



The reader will perceive that these series essentially differ from those in 

 the ' Fundamenta Nova.' The series given in the ' Fundamenta ' may be easily 

 deduced from formulae (1). • . -(10). As, however, my object is to exhibit 

 the progress which the theory of elliptic functions has recently made, I care- 

 fully abstain from writing down any series which are given in Jacobi's great 

 work. 



Section 11. — AVe now return to theorem A of last section. Transforming 

 the series in the second member of this equation by means of expression (1) 

 in last section : since 



whence 



1 



and similarly for 



S 



we obtain 





-'"sin(a;+s»?) 



^jg2s5» 



_=0;(o)6(-«)'Mf^^£±^), 



'sin(a-.v+s»'0 tii{a,-x)i)yh 



qS^2sU 



and S° 



tSc^sSi 



2*e 



'-"» sin(/3— .r+sj'i) -°° sin (y— aj+sj-i)' 



' ^ ^ 0iO3-a)0i(y-a) 0i(a —r) 



e,(\-y)6,(fx.~y)d,(p-y) d,(y-x + B) 

 0i(a-y)0i(/3-7) ■ 0,{y-x) 



Tor .r=0 we have 



e.Xe^fid.p _0 ,(a+3) ^,i\-cc)^X^^-cc)6^(p-a.) 

 0,a0,/3 0,7 «iO ' 0,a 0j(/3-a) 0,7— a 



fl,(/3 + g) e, a-/3)e,(/i-/3)e,(p-/5) 



0,a •"0,/3 0,(a-/3,)0,(y-/5) 

 , 9,(y + g) ^,(\-y) (^l(^l -y)Ol(p-y) 



6,S • e,yd,{a-y)d,{l3-y) ' 



> 



(1) 



