262 DEVELOPMENT of a certain 



Taking now the fluents, when | and <p — t, and fubftituting 



wtflA. for / . 771, a ^° 7 * a ~B f° r /7 ~> 



we get, I \ v/i — e a fin 2 ^ = (i — e l ) ra'A — i 6 ( i — 6 *) *&B. 



Let A^/B be another ellipfe, having its femi-tranfverfe alfo 

 zr i, but its excentricity =r 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 Kb be denoted by ^, the fluxion of 

 the elliptic arch <i/"will be exprefled by \^/i — e'fin^j there- 

 fore, when { = r, we have /^/i — e l fin a >p equal to AdB, half 



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

 meter of this fecond ellipfe, and our laft equation becomes 



(E') = (i -— r)^ 5 A — 7«(i — e*)va 3 B, 

 and, by proper reduction, and fubflitution of - for s, we finally 



get 



■r — 2fl A — 2 v (E ') . 



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 firfl coefficients of the feries, and upon which all the re- 

 maining coefficients depend, to the rectification of the ellipfe : 

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

 of infinite feries, in every cafe that can poflibly occur. 



io. I next obferve, that in determining the coefficients A, B, 

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

 ted ; for, inftead of them, we may fubftitute 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 



