278 DEVELOPMENT of a certain 



approach very fad to the f atio of equality, which is their limit i 

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

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

 other. 



Let us denote the limit, to which thefe arches continually 

 approach, by 6, and it will prefently appear, that the fame 



arch Q is alfo the limit of the feries of fluents / ^ . 



r ^' r ^" &c. or of P, P', P'", &c. fo tliat 



we have ultimately 



P = ^(i+0(i+^')(t+O>&c. 



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



/" «<""»'* r ^'fin'iV /Z i^fin'4^'^ gjc- or N 



/,—<Mm'p'V'/i—£'' fin '**■'» y/i — /■"fm'4f"^* ' 



N', N'', &c. continually approaches to j6 fin '«^, where « de- 

 notes a number indefinitely gr?at ; but in this cafe y/ fin *«tf 

 = /'/(- — - cof 2»^) is equal to -,afinite quantity; now if we re- 

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

 tiplied by the two infinite produifls e'.e'.e'". &c. and ^-^ • ^-^ • --^, 

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



the quantity e e" e\ &C. X , . ^ / a — ^ T ^« 'o ^e 



reje<5\ed. 



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



/Li!l»^,it is evidently equal to ^r { f \ ^ -/^ v/IZ^SH^ } i 



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

 e ■=. the excentricity, and ^ = 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 



thae 



