396 Roijal Societij. 



in which the double sign denotes two terms, which must hotli be in 

 eluded. But besides, if the symbol (p(r) stand for 



[(l_;.-=)(lJ,.^)(l_,/>)...&c.]->, 

 A/a.^0-).O(7,'rH + |x)=r»=.(l + H±2«)(l + ?-3±2«)(i+,.5±2„). 



A/«.^(r).A(</,7rH + ^7r) = r"=.(I-7-i±2»)(l_,.3±2„)(i_,.5±2„)., 



From these formulne not only all of Gudermann's developments 

 for calculating elliptic integrals in every case are deducible, but 

 others also, it seems, of a remarkable aspect, in the difficult case of 

 q and c being extremely near to 1 . 



We produce the two which seem to be simplest. Let B be to 6 



what C is to c, and Tan x represent ^^ , where 2 Siu x stands for 



Cos a; 



e»_g-^ and 2 Cos a; for £•*'+£"■''. Then when c is very near to 1, 



we compute G and thereby E from the series, 



B.G(e,.)=--^^-(l-Tan3 + (l-Tan--=^-) 



_ ?1 -Tau l±f^ + fi -Tan ^2!^:^) 



_A-Tan^') + [l-Tan^^]-&c.... 



The third elHptic integral is in the same case deduced from a series 

 of the form 



!LZ:^ •i-jj'-tan-Ytani.Tan^") j + ji-tan-'Aan^.Tan^H 



- |>-tan-iCtan>.Tan^!±£)} + lj-t!in-^ftmj.Tm?^)\ 



— &c. ... 



Finally, the essay developes above thirty series which rise out of 

 this theory, nearly all of which are believed to be new. The most 

 elegant of them may find a place here. Writing, for conciseness, C 

 so related to c that F(c, |7r) = |7rC, and .-. F(c w)=Ca^, we have 



^ . sin 2x , , sin 4x , , sin Gx , „ 



\^a). to=x+^ ■+hr^ — T~+^-n — ^— + ^^'=- 



^ ^ Cos na Cos iTt-a Cos Sttci 



/ 7 N /-. . / N , , 2 cos 2a; , 2 cos 4.r , 2 cos 6x , n 



(b). CA(c, w)=l + -^^^ + ,, ^ +^ — - — + &c., 



^ ■' ^ Cos ira Cos 27ra Cos 3ira 



i- \ 1 /-.2 2 • o sin 2x , 2 sin 4a? ,3 sin 6x . „ 



(c). ACVsin2w=- f-7^ — ^ +?T-^ — + &c.. 



^ ■^ ° CosTTfl Cos27ra Cos37ra 



/ ,. -.J/ . 1— cos 2a;, , 1— cos 4a?, i I— cos 6a; , „ 



(d). Y(c,w)= — : +1 — ^-- +i--^^^ + &c... 



^ ^ sm Tra sm 2na sm 3na 



/ \ 1 /-. n / \ sin 2a; , sin 4a; , sin 6x „ 

 ^ ' sm 7ra sui Ittci sin ona 



This is virtually eq. 49 of Legendre's Second Supplement, § 7. 

 In eq. 53 of the same, he has a develo])ment of sin" w, which is given 

 by Mr. Newman in a notation similar to eq. (c) above. 



