ALGEBRAIC FORMULA. 269 



15. The method of determining the coefficients, which is de- 

 rived from this laft feries for the redification of the ellipfe, 

 therefore, feems better than fuch as would be afforded by either 

 of the other two feries given in this paper. For, befides conii- 

 dering that there is no quantity required for the computation of 



— , which was not pre vioufly wanted in the computation of 



-, it appears that the logarithms of the quantities e\ e\ &c. alfo 



I + f', I + e", &c. may be all derived from e, and from each 

 other,, with great facility, by means of the common trigonome- 

 trical tables, in the following manner. 



Let a be fuch an arch that fin « — ^, then cof a, =r s/x — e^, 



and confequently e' - ^-'/J=7^ _ l^^ = tan-, alfo i -i-c 



zr. fee -. Again, by taking a fuch that fin a iz e\ we find, in 



like manner, f* = tan — , and i-\-e"=z fee — ; and fo on we may 

 go, as far as may be neceffary. Hence it follows, that 



P = fee — . fee — . fee — , &e. 

 a a 2 ' 



^-^ fin « fin a . fin a' , fin a . fin a' . fin a* , o 

 Q^= — + a ■ z + z . » ■ a + ^'C. 



and fince we are provided with the logarithms of fee -, fee - 



&c. alfo fin a, fin a', &e. the arithmetical calculation of P and 

 Q^will be very eafy. 



I MUST obferve that this lafl method of determining the co- 

 efficients A and B, in the cafe of the exponent « =: — -, by means 

 of the infinite produd P = (i -f f') (i + "^'O (i + ^"') &c. and 

 the fea-ies Orr ^ -f 7^, +ttt + ^<^' coincides in the refult with 

 the' method given by Mr Ivory, for determining the fame co- 

 efficients, when the exponent is — -, in his very ingenious pa- 



VoL. v.— P. II. M m per 



