2^2 DEVELOP MENT of a certain 



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

 caunot, however, be efle*ft;cd by formulse 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 formula for their redlification, are more 

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

 cle and parabola. But it is to be obierved, that whatever difii- 

 cuky thei'e may be in the redlification of the ellipfe and hyper- 

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

 the moll 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 redlification of two elliptic arches *. Thus, it ap- 

 pears, that whatever facility the following feries may afford for 

 the re<flification of the ellipfe, it muft alfo be underfl:ood to ex- 

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

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

 thefe curves is neceflary. 



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

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

 each other, that 



fin 20 /• , 



where ^ denotes an invariable qviantity. 



Then it is evident, that when <p zz o, cp' is alfo =: 0, and 



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



from o to -. 



Taking now the fluxion of our afl"umed equation, we get, 



^ r v~i ! — ?• col 2®; 



v''! + e" + Zf* col 2?i ' (i +f-- + ac'cof 2^)^ 



But 



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

 lofophical Tranfaftions for 1775, and afterwards in his Mathematical Memoirs. 



