ALGEBRAIC FORMULA. 



279 



P-* 



that <pv/i — • <?* fin *p expreffes the fluxion of the elliptic arch 

 FD j let us denote that arch by z, and we have N = -~- ; fo 

 that upon the whole we find 



+ (i + -i + -£ll + £^ + &c.)P 



and from this equation trie manner of exprefling z, by means of 

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



11. I shall now colle& 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 1 = AC the femi-tranf- 

 verfe axis, 

 e =: the excentricity, 

 z = DF any arch of the 

 ellipfe, reckoned 

 from the extremi- 

 ty of the conju- L- 

 gate axis. 

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

 between the conjugate axis and FG, an ordinate to 

 the tranfverfe. 

 Compute the quantities <?', e'\ e'", &c. fin 2<p', fin a$\, fin Sep'", 

 &c. from thefe two feries of equations, 



.,__i-Vi-<" 



fifJ.T {m2<P 



L - ,+•,_,••' 



nn2? (i+ov; 1 -^^^ 



vr _ I _ v/l _^ 



fi n .-,//_ *n 4 *' 



- 1+v /, _«*»'» 



111149 " (i+/0v^-^fin*V* 



ff I-l't— *' 



fi n fl>- fm8 ^ 



I — y'l — e'"* ' 



J.1V — r 



(i + «'") / 1 — e* s fin * 4^ ' 

 £ -i/^/^tv £111160'" 



Na2 The 





