'ALGEBRAIC FORMULA. 



279 



P — ! 



that <pv/i — ^' fin '<p expreffes the fluxion of the elliptic arch 

 FD J let us denote that areh by z, and we have N =: 

 that upon the whole we find 



fo 



P — « I ' \a ' 2.J '22.1 ' i.i.t.z 



2 ^ ^^) P 



+^0 



fi„4^+(2S!±2fet^fcK3,„8,.,+8,^. 



and from this equation tHe manner of exprefllng z, by means of 

 the infinite produdl P, and two infinite feries, is obvious. 



1 1. I SHALL now coUedl into one point of view the refult of 

 the preceding analyfis, in the form of a pradical rule for finding 

 the length of any arch of an ellipfe. 

 Let I = AC the femi-tranf^ 

 verfe axis, 

 e =: the excentricity, 

 z = DF any arch of the 

 ellipfe, reckoned 

 from the extremi- 

 ty of the conju- 

 gate axis. 



<p = HK, an arch of the circumfcribing circle, intercepted 

 between the conjugate axis and FG, an ordinate to 

 the tranfverfe. 

 Compote the quantities e', ^, e"', &c. fiii 2<p'\ fin ^<p\ fin 8<p"', 

 &c. from thefe two feries of equations, 



fin 2ip'' = 



<f"' = 

 &c. 



fin 2? 



fin 4<p'^ zz 

 fin8(p"' = . 



(i + O v/i— f*fmV'^ 

 (in 4*' 



fin 8?>" 



fin 16^'' =r^ ^ni6p-' 



&C. 



N tt 2 The 



