212 A NEW and UNIVERSAL SOLUTION 



Now, fince fin (w — r) r: s fino- X cof w, it is evident that 

 fin (« — !r) can never be greater than ^. Alfo, if § = a tan ^ , 



wc fliall have fin g L a tan ^ ; therefore ^ i^ cof p L a : 



^ ^ tan^ ^ 



therefore, fince a -n ^J i — ^, it is is manifeft that fin j is great- 

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



arch ^, determined by the formula f = « tan ^. 



It now remains that -we determine the maximum value of 

 the function ^ — a tan ^ : for this purpofe, let j' = tan §, and 



fince e =^ — r — » we fliall have, by the ufual method, 



1 



whence J/ — tan ^ — \/ (- — ij 



Therefore, if we take tan^ ■=. *// ^ _^^ ^, the arch 



4 



y — ff can never be greater than ^ — « tan ^. 



If we take £ =: i, we fliall have tan ^ zz V (^ — ^)' whence 

 ^ r: 2i°28' 14" nearly, and ^ — -^ tan g zi .03411, which is 

 the length of an arch of i " 57' nearly. It is therefore certain, 

 that, even in the extreme cafe, when s n i, the arch t cannot 

 differ from half the arch of eccentric anomaly, more than i "57' ; 

 a very fmall error, confidering that the firfl fuppofition of « rr k 

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

 the feries 5r, -x , -k' , &c. converges to the true length of half the 

 arch of eccentric anomaly with uncommon rapidity. 



g. We have now fliewn, that, by means of the finite equa- 

 tion fin {n — v) — f fin V X cof v, together with the foi'mula 



^ ::: s X ^^ -, we may deduce a feries of arches, converging 



n — V 



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



fonings 



