268 DEVELOPMENT of a certain 



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

 new; and which, hcfides poffefliug the fingular advantage of 

 converging rapidly in every cafe of excentriclty that can pof- 

 fibly occur, has other properties that render it peculiarly well 

 fitted to the p-urpofe 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' = 1^-^, /^ = '-"^lEIl^ , e'" = i=^^^;, &c. 

 Compute P = (I + O (i -fO (i + O &c. 



''ind a= - + - + ;xr + i:^x-2 + &c. 



Half the perimeter = 5rP(i -— eQ_). 



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", &c. are 

 the fame fundions of e, that /', e'", &c. ai-e of e'. Hence it 

 follows, that if 



F = (i + O (i -h O (i + ^"). &c- 



■. weget, E' = ^P'(i— ^'Ql). 



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



Ql, it appears that P' = -^., and 01= '^- i =^^^Qz- 1, 



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 die coefficients A and B) may be expreffed thus : 



5 = p(i-.(^). 



£ = P(i_i(x-|-/)(^). 



15. The 



