26o DEVELOPMENT of a certain 



Let us now fubftitute 2 cof-| — i, for its equal, cof(p, and 



V,.' 



the fluxionary exprefllon ^v/i + a* — 2e cof (p becomes 



( I + s) <p^/ 1 — IT+ly C°f ~f ; '^o that if we aflume e fuch, that e^ = 



^^^„ we -alfo get 



'^^'A = ^j-p-yi-— y ^V I — ^' cof ^. 



Let ADB be an ellipfe, of which the femi-tranfverfe axis 

 AC=: I, and excentriclty = e, and let AKB be a circle, having 

 AB, the tranfverfe axis, for its diameter ; let FG, any ordinate 

 to the tranfverfe axis, meet the circle in H ; then, if we put the 



arch AH zz -, it is well known 



that -<pv I — ^' cof— will de- 

 note the fluxion of the ellip- 

 tic arch AF j therefore, when 

 <p =. -jr, the fluent of 



(ps/ I — f' cof —'will evident- 

 ly be exprefled-by ADB, the femi-perimeter of the ellipfe : Let 

 us denote the femi-perimeter by (E) and we have a-vA zz 



-. — ■ ^f -- — ^T ; hence, and by fubflituting - for s, we get 



A = - 



(E) 



(rt + tf) (a ■ 



and becaufe e, the excentriclty of the ellipfe, has been afTumed 



2\/t 



i\/ab 



and the 



— ^^, and e =, - ; therefore the excentricity —-^-^jt^, 



a — h 



femi-conjugate axis _ -;;^- 



9. Having now found a geometrical exprefllon, fufiiciently 

 fmiplc, for A, I proceed to invefligate a fimilar expreflion for 



the 



K 



