I 35^ 1 



The fecond power of the eccentricity or two terms of the fcries 

 will be fufficient for the orbits of the Earth and Venus. The third 

 power of the eccentricity or three terms for Jupiter, Saturn, and the 

 Georgium Sidus, and four for Mars. But fix or feven are neceffary 

 for Mercury. It is more tedious than difficult to continue this feries 

 to a greater number of terms. The above folution of this ufeful and 

 celebrated problem, befides being dired is greatly fhorter than any 

 before given : Even than the method of Cagnioli, given by De La 

 Lande, in the third volume of his Aftronomy, edition 1792, where 

 the feries is continued to the ninth power of the eccentricity. By a very 

 ingenious artifice there given the folution by indeterminate coeffici- 

 ents is very confiderably fhortened. The legality of that artifice might 

 however be juftly doubted, and the truth of the conclufion de- 

 duced fufpeded, unlefs verified by other methods. 



Example IX. From the equation 



cx''x-\-jyx —ay to find j' by a feries afcending by the powers 

 of X, n being a whole pofitive number (Simpfon's Fluxions, Vol. IL 



293)- 



SoLUTiox. When x = 0, let j* = Y. Then taking the fluxions 

 of the given equation when x = 0, and x flows uniformly. 



la. yx = ay 



_, . 3. 

 2^, yx = ay 



tA 



