260 DEVELOPMENT 1 of a certain 



Let us now fubftitute 2 cof — — 1, for its equal, cof <p, and 

 the fluxionary expreffion <p^/i -{- e 7 - — 2e cof <p becomes 

 (1 + s) (pif 1 — JTJTy cof -|; fothatifweaffume^fuchjthat^z: 

 (T+o 1 * we-alfoget 



^ 3A = jj+7)zr=7yf M l ~ f2 cof ■?• 



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

 •AC = -i, and excentricity — <f, 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 j then, if we put the 



r 



arch AH — -, it is well known 



that ^<pV 1 —^cof-^ will de- 

 note the fluxion of the ellip- 

 tic arch AF ; therefore, when 

 <p =z *, the fluent of 



(p^J 1 — e z cof — c will evident- 

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

 -us denote the femi-perimeter by (E) and we have #-VA — 



flf — ^r ; hence, and bv fubftituting - for «, we get 



A ___!_ (E) 



A ~ (_u+b)(a — by X 7T ; 



and becaufe <f, the excentricity of the ellipfe, has been affumed 

 ~^-,and g — - ; therefore the excentricity — a + b > and the 



a — b 



femi- conjugate axis — -^-^. 



9. Having now found a geometrical expreffion, fufEciently 

 fimple, for A, I proceed to inveftigate a fimilar expreffion for 



the 



