232 A NEW and- UNIVERSAL SOLUTION 



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

 mation, derived from the afTumption e — e, is almoft of no ac- 

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



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



p 



fion be true of the more exact afTumption e =z s . 



r 



Let 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 1 - ; write fin <p for x, and cof l <p for I — x~- ; and 

 we fhall eafily obtain, 



fin <p zz fin m -f- e 1 - fin <p cof 2 <?. 

 "In this formula it is clear, that the term <r- fin <p cof 1 <p is incon- 

 fiderable in comparifon of the other two : therefore fin <p rr fin m 

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

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

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

 very fafl to the exact value of <p ; viz. 



fin <p — fin m -f- e 1 fin m cof 2 m 

 fin q>" — fin m -f- e 1 fin q>' cof 1 <p 

 fin 4" — fia m + e* fin 9" cof 2 9", 

 and fo on. The error of the firfl approximation ?', is of the 



order 



/ this value of fin /* in the feries for e, we have, (neglecting the terms above the 

 order e 4 ), 



3 4- 



e = e fin * m + £ — fin m X fin 2m ; 



24 24 



and this value of e is exaft, as far as the order e 4 inclufively. 



But the afTumption e = e x , where t = , being thrown into a 



feries, we get, 



c = £ -fin 1 to 4- ^--- fin A m-~- &.c. 



24 640 



a.nd therefore the error of this afTumption is of the order e 4 . 



