232 A NEW and- UNIVERSAL SOLUTION 



It has been fhewn above, that the error of the firft approxi- 

 mation, derived from the afTumption ^ =: s, is almoft of no ac- 

 count, as to any real pradlical purpofe, in the orbits of all the 

 planets, excepting Mercury : and much more will this conclu- 



fion be true of the more exact ailumption e zz i . 



r 



Lf.T us now confider the cubic equation which the rule re- 

 quires to be refolved: the equation is 



and, in the cafe we are now occupied with, e is fmall, being 

 nearly equal to the eccentricity. Multiply all the terms of the 

 equation by e^ ; write fin (p for at, and cof ' (p for i — x- \ and 

 we fliall eafily obtain, 



fin ip =: fin in -\- e"- fin p cof - <!>. 

 "In this formula it is clear, that the tei-m e- fin (p cof" <? is incon- 

 fiderable in comparifon of the other two : therefore fin ? zz fin Jii 

 nearly ; and, confequently, the two arches <f and m will differ 

 but little from one another. From this confideration, we readi- 

 ly derive a feries of approximations, <p\ <f" , p", &c. converging 

 very fall to the exa6l value of ?> ; viz. 



fin <!>' zz fin m -j- e- fin m cof* vi 



fin (f" zz. {\x\. m -\- r fin ?>' cof* ^' 



fin <p" = fin m -f- e'^ fin ?>" cof* ?", 

 and fo on. The error of the firfi: approximation ?>', is of the 



oi'der 



' this value of fin /i in the feries for e, we have, (neglefting the terms above the 

 order £*), 



« ^ E fin ' m + — fin TO X fin 2CT ; 



24 24 



and this value of e is exaft, as far as the order e* inclufively. 



But the affumption e = EX , where t = , being thrown into a 



feries, we get, 



f ' e' 



f = E fin*«!4- ;^- fin ■• » —f- &c. 



24 040 



and therefore the error of this affumption is of the order £•*. 



