2^2 DEVELOP MEN? of a certain 



2. The indefinite rectification of the ellipfe, or hyperbola, 

 cannot, however, be effected by formulae of fuch fimplicity as 

 thofe which exprefs indefinite arches of the two former curves ; 

 for the algebraic equations which define their nature, and from 

 which we derive the formulae for their rectification, are more 

 complex than the equations which define the nature of the cir- 

 cle and parabola. But it is to be obferved, that whatever diffi- 

 culty there may be in the rectification of the ellipfe and hyper- 

 bola, it is now confined entirely to the ellipfe ; for by one of 

 the moil happy applications that has ever been made of the mo- 

 dern analyfis to geometry, it has been difcovered, that the rec- 

 tification of any hyperbolic arch whatever may always be re- 

 duced to the rectification of two elliptic arches *. Thus, it ap- 

 pears, that whatever facility the following feries may afford for 

 the rectification of the ellipfe, it mufl alfo be underftood to ex- 

 tend the fame to the rectification of the hyperbola, and to every 

 problem, for the folution of which the rectification of either of 

 thefe curves is neceffary. 



3. But to proceed with the inveftigation of our formula, let 

 ois fuppofe that <p and <p' are two arches of a circle, fo related to 

 each other, that 



fin 2<9 r , 



v/i +e /l + ^e'col2p r 



where ef denotes an invariable quantity. 



Then it is evident, that when <p zz o, <p' is alfo zz o, and 



while <p increaf's from o to a quadrant or -, <p' will alfo increafe 



from o to -. 



2 



Taking now the fluxion of our affumed equation, we get, 



y , r A — i =(p Cof 2 tp; 



v/l + « /z H-2<?'cof2p A ( x +e ~- +2e'co£2<p)i 



But 



* This difcovery was made by Mr Landen, who publifhed it firft in the Phi- 

 lofophical Tranfa&ions for 1775, and afterwards in his Mathematical Memoirs. 



