i8o RECTIFICATION of the ELLIPSIS, &c. 



I THUS obtain <p V i — g^ cof^cp — 9 / ^^ + /^* — 2ab cof 2(p: 

 and, taking the whole fluent, while (p from o becomes =: w, it is 

 manifeft, from what has been premifed, that the femiperiphery 

 of the ellipfis is ir 



or puttmg —zne =. > and ^ r: ——7 zr % zz -— - 



the femiperiphery of the ellipfis ~ - ^ X 



(■ + "-^=+."^''+ ^^^^°+&c.) 



In this feries, as was before obferved, ^ is a fmall fradioii 

 even when s is very confiderable, and the coefEcients arc 

 more fimple in the law of progreffion, and converge fafter, (ef- 

 pecially in the firfl terms), than in the common feries. 



If we fuppofe the ellipfis to be infinitely flattened, in which 

 cafe g n I, and ^ — i, and the femiperiphery zz 2, this feries 



gives 2 z^ ^ xfi + ^:+-i^i;+^^;^i;^ + &c.),andfo 



But, w^e may remark, that as we have here obtained the fum 

 of the fquares of the coefficients of the binomial when the ex- 

 ponent is 7 ; fo, from the fame fource, we may determine- the 

 fum of the fc]^uares of the coefficients correfpondingto any other 

 exponent, at leaft by a fluent. 



For taking the whole fluent when (^ — sr, we have- 



J{n^ + ^^ - 2^3cof (pj^^z. ^- ^ {^+^'S +^^- .7^+ y'' 5-+^^-) 

 and fo when azzii, and^ r: 1, 



Now,, 



