Ill 



The author fliews by feveral examples that in orbits of great and 

 fmall excentricity the firft approximation is always fufficient to apply 

 the fccond rule with advantage. 



This will be feen generally by the following eonfiderations by which 



a limit of the error of the firfl approximation is obtained for any value 



of c. 



The two feries above given each converge floweft for a given value 



c 

 of c when n is leaft, becaufe then s (=j-, — ) is greateft. The terms 



?i 



omitted therefore bear a greater proportion to the terms retained, and 

 confequently the error of the value of s determined from the cu- 

 bic equation is then greateft for a given value of c. We derive then 

 the general limit of the error of any value of c, by taking fzri and 



therefore «=y/io. The equation becomes »z=:y/io x : — s + — + &c. 



6 56 

 Now it will eafily appear by confidering the formation of the co-efEcients 



J' 



of the two feries in the cafeof «=v'iothat the terms f-&c. are all pofi- 



. ' . . 56 



tive and their co-efHcients convergmg. Hence the fum of the terms 



/' . . ' ^' 



— +&C. omitted is lefs than — X, But if the cubic equation 



56 s^ i—s^ 



JO J \/io j' 



»i=v'iox — s be varied by adding — x to the right hand fide, the 



6 56 I J2 



is' 

 variation of J will be nearly^— — x an d therefore of 



280 I— j' 



I J' « v'lo ' ^ 



M«= - — X X — Xi X /, a. This quantity then will be al- 



280 I — s'^ cs, a 280 



, ways 



