178 RECTlFICjriON of the ELLIPSIS, Sic. 



Let e denote the excentricity of an ellipfe, of which the femi- 

 tranfverfe axis is unity, and in the length of the femicircle, ra- 

 dius being unity ; Then, 



if we put e — — ; — . ,- ) 



half the periphery of the ellipfis will be 



the coefficients being the fquai-es of the coefficients of the radi- 

 cal / I — e\ 



The common feries Is, 



, 1. r 1.1 1-3 I-I-3 I-3S , „ V 



arXfl— -•- f' — — • — 6"— ■ £*— &c.). 



^ 22 2.4 2.4 2.4.6 2.4.6 ' 



The fir ft of thefe feries converges fafter than the other on two 

 accounts : firft, becaufe the coefficients decreafe more rapidly ; 

 and, next, becaufe e is very fmall in comparifon of £, even 



•when £ is great : Thus, if e be ^, <? will be -•, and ^* zz ^. 



In order to point out the way in which the preceding 

 feries was difcovered, let us fuppofe [a^ -\- b'^ — 2ab cof (p)" 

 = A +, B cof ip + C cof 2(p -f &c. ; and to determine the 

 coefficients. A, B, C, &c. let us, with M. de la Grange, 

 confider the quantity [a'' -{- b'^ — 2ab cof (p) as the produdl of 



the two imaginary expreffions ia — bc'^'^ ~^Y and 

 \a — ^<r~^^ ~'j, where c denotes the number whofe hy- 

 perboUc logarithm is unity. Then, by expanding the powers 

 {a — be ^^ -■")", 2ind {a — bc-'^^-')'' into the fe- 

 ries «"( i —« .i-c ^/ -' -|- /3^ ^^ /-^ _ ^^3?*/- ' + &c.) 



and 



