278 DEVELOPMENT of a certain 



approach very fad to the ratio of equality, which is their limit j 

 and the fame will alfo be true of the ratios which every two 

 adjoining arches of the feries <p, <p\ <p", <p'", &c have to each 

 other. 



Let us denote the limit, to which thefe arches continually 

 approach, by 0, and it will prefently appear, that the fame 



arch 6 is alfo the limit of the ferie3 of fluents / f . 



C * f ** &c. or of P, P, Pf, &c. fo ttiat 



J t/ i— e'Hm'z^ J 1/1—cr- fm*ty* 



we have ultimately 



P = 4 (i +0 (i + (t + A &*• 



9. In the fame way it appears that the feries of fluent* 



/ "* gfin»» / ~ yfin'i'g' /] g*fin» 4 p* & c . or N 



/. -(Min'f'V /i — *' 1 fm ~' y y^i — t* 1 fin Mf^ 



N', N', &c. continually approaches to yd fin *»^, where » de- 

 notes a number indefinitely great j but in this cafey & fin *n9 

 — /i*(- — r cof 2»$) is equal to -,afinite quantity; now if we re- 

 mark that this quantity, which enters the value of N, (Art. 6.) is mul- 

 tiplied by the two infinite produds e\e*.e'". &c. and —• • —■ • i££j 

 &c. both of which are evidently equal to o, it appears that 



the quantity * *"*", &c. X > a ^ v a J v a — * X - is to be 



rejected. 



10. Finally, with regard to the quantity N itfelf, or 



now if we put 1 = AC, the femi-tranfverfe axis of an ellipfe, 

 * = the excentricity, and p = HK, any arch of the circum- 

 fcribing circle, intercepted between the conjugate axis, and 

 FG an ordinate to the tranfverfe axis, it is well known 



that* 



