i8o RECTIFICATION of the ELLIPSIS, &c. 



I THUS obtain ^ / I - £^ coP<p = ^ ^ a^ + b^ — 2ab cof 29: 

 and, taking the whole fluent, while <p from o becomes = ar, it is 

 manifeft, from what has been premifed, that the femiperiphery 

 of the ellipfis is = 



or ptittmg ~=e = , + ^,^,^ ' ^"'^ '^ - TT-*:^ - —& - i + -' 

 the femiperiphery of the ellipfis zz Yqr7 ^ 



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

 even when s is very confiderable, and the coefficients ai'c 

 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 £ = I, and ^ =:. i, and the femiperiphery =. 2, this feries 



gives 2 = lx(i + ~+ ^ + '^^^ + &C-.), and fo 



V- ' +'^ + 4^4' ^ 2^4^o» + 2.\4'.o'.b' + ^^• 

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

 of the fquares of the coefiicients of the binomial when the ex- 

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

 fum of the fquares of the coefiicients correfponding to any other 

 exponent, at leaft by a fluent. 



For taking the whole fluent when (p — ar, we have- 



j(^^ + b^-—^abco{(p)%-a" z, (i + c.\~ -j-(B\ ^^+ y\ ^-f &c. ) 



and fo when « = 1, andA := 1, 



j ^^ + ¥ — iabco{(p J^_^ ^ ^, ^ ^,_}. y, ^ &^.,. 



Now, 



