178 RECTlFICjriON of the ELLIPSIS, Szc. 



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

 tranfverfe axis is unity, and w 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 fquares of the coefficients of the radi- 

 cal V \ 6*. 



The common feries is, 

 jjrxfl .— £^ — — g^— . g*^ — &c.). 



^ 2 2 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 e, even 



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



In order to point out the way in which the preceding 

 feries was difcovered, let us fuppofe [a?- + 3* — lah cof (p)' 

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

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

 confider the quantity [aP- -\-h'' — lah cof ip) as the produdl of 



*he two imaginary expreffions [a — bc'^'^ ~~'^\ and 

 [a — ^<7""^^"~^j, where c denotes the number whofe hy- 

 perbolic logarithm is unity. Then, by expanding the powers , 

 [a^hcf^-^)\^v,A [a^bc-^^-'f into the fe- 

 nes^"(i — «.A^^/-i + ^^2^/-i_y^3?/-i 4_ &c.) 



and 



