268 DEVELOPMENT of a certain 



I shall now give a third feries, which, fo far as I know, is 

 new; and which, befides poffeffing the fingular advantage of 

 converging rapidly in every cafe of excentricity that can pof- 

 fibly occur, has other properties that render it peculiarly well 

 fitted to the purpofe for which it is wanted in this paper. 



Let i denote the femi-tranfverfe axis, and e the excentricity 

 of an ellipfe, as before. 

 Find e> - IzgjgEg fy _ i-VIw^ e „, _ i-/^ & 



Compute P = (i + <?') (i +/) (i -f- e"') &c. 



A r\ e I ee ' L et ' e " I " W " r c 



and Q^= - + - + — + ^rr 2 + &c. 

 Half the perimeter rr dP(r — <?Q_). 



It is evident from the form of this laft- feries, that if e be con~ 

 fidered as the excentricity of E, one of the ellipfes by which we 

 have expreffed the coefficients A, B, (Art. 13.) it prefently fol- 

 lows, that e\ the next quantity that occurs in the feries, will de- 

 note the excentricity of E', the other ellipfe. Now e\ e a , &c. are 

 the fame functions of e, that /, e'" f &o are of e\ Hence it 

 follows, that if 



F = (1 -f O (1 -f e'") (1 + <?"), &c 



e > gigii e ' t " e "i 

 Q1-— a + 2^ + 1X7 + &c * 



xveget, E' = *F(i — e'QT). 

 But in comparing together the values of P and P', alfo Q^and 

 Ql, it appears that P' ± ^ and Ql= a £- 1 xx&^&Qm ,, 



fo that the ratios of the ellipfes E and E' to their common cir- 

 cumfcribing circle, (which are what we want in the computa.- 

 tion of the coefficients A and B) may be expreffed thus : 



f = P(i-*Q> 



15. The 



