ai2 A NEW and UNIVERSAL SOLUTION 



Now, fince fin Qi — r) zz s fin k X cof T, it is evident that 



fin (n — x) can never be greater than -. Alfo, if g — a tan g> 



«- 

 we fhall have fin p L a tan g • therefore -^ zz cof p L a : 



tang ? 



therefore, fince a = sj 1 ■ — B ~, it is is manifeft that fin g is great- 

 er than ;. Whence it is obvious, that n — n is lefs than the 

 arch £, determined by the formula g zz a tan ?, 



It now remains that we determine the maximum value of 

 the function g — a tan g : for this purpofe, let y zr tan g> and 



fince g — J_ |t , we fhall have, by the ufual method, 



whence y zz tan g zz \/ [ ij. 



— rt~o; 



/ i 

 Therefore, if we take tan g — v v^TTZT » tne arcn 



4 



v — ^ can never be greater than g — a tan g. 



If we take g zz i, we lhall have tan g zz v (ttz — i)> whence 



^ — 2i°28' 14" nearly, and g> — « tan g> zr .03411, which is 



the length of an arch of 1 ° 57' nearly. It is therefore certain, 

 that, even in the extreme cafe, when sz i, the arch ar cannot 

 differ from half the arch of eccentric anomaly, more than 1 °$j' ; 

 a very fmall error, confidering that the firfl fuppoiition of n zz v 

 is very wide of the truth. We may therefore conclude, that 

 the feries t, «■', w", &c. converges to the true length of half the 

 arch of eccentric anomaly with uncommon rapidity. 



9. We have now fhewn, that, by means of the finite equa- 

 tion fin {n — v) zz e fin v X cof e, together with the formula 



e zz g X -, we may deduce a feries of arches, converging 



very rapidly to half the arch of eccentric anomaly. The rea- 



fonings 



