262 DEVELOPMENT of a certain 



Taking now the fluents, when \ and <? = «■, and fubftituting 



we get, Av/i — e^fin'vf-'r: (i — s") ^^'ir'A — i s( i _ e^) ^^53. 



Let AiB be another ellipfe, having its femi-tranfverfe alfo 

 =r I, but its excentricity ^ e ; let its conjugate axis meet the cir- 

 cle in K, and let gf^ an ordinate to the tranfverfe axis, meet the 

 circle in h ; then, if the arch YJi be denoted by ^j', the fluxion of 

 the elliptic arch ^/will be exprefled by jv/i — £'fm^v|'; there- 

 fore, when vj/ =: ff, we have /vf'/i — e'fin'vp equal to A^B, half 



the perimeter of the ellipfe. Let us put (E') for the femi-peri- 

 metcr of this fecond ellipfe, and our lafl; equation becomes 



(£') = (i — £') ff«' A — ^ e(i — eO'T^^B, 



and, by proper redudlion, and fubftitution of- for £, we finally 



get 



R -Hf A— — i— -X ^^^ 



and, fince the excentricity of this other ellipfe is -> its femi-con- 



jugate axis = ^ . 



Thus we have reduced the determination of A and B, the 

 two firft coefficients of the feries, and upon which all the re- 

 maining coefficients depend, to the redlification of the ellipfe : 

 now this is a problem which we can readily refolve, by means 

 of infinite feries, in every cafe that can pofllbly occur- 



10. I NEXT obferve, that in determining the coefficients A, B, 

 &c. we are not confined to the two ellipfes jufl:now inveftiga- 

 ted ; for, inftead of them, we may fubfl:itute other two, having 

 their excentricities as great or as fmall as we pleafe. This pe- 

 culiarity of our folution depends upon a very curious relation 



which 



