i82 RECriFICAriON of the ELLIPSIS, ^c. 



feries of this form, A-f- B co£<p -\- C cof2(p + &c. The quan- 

 tities a and b reprefent the diftances of the difturbing planets 

 from the fun ; and when thefe bear fo great a proportion to one 

 another, (as in the cafe of Jupiter and Saturn, or Venus and the 



Earth), that the fracflion - is large, it becomes extremely difficult 



to compute the coefficients A, B, &c. by feries, on account of 

 the great number of terms that muft be taken in. This matter 

 not a little perplexed the firft geometers who confidered this 

 fubjedl, and they were obliged to approximate to the quantities 

 fought by the method of quadratures, and by other artifices. 



Two things are to be attended to with regard to the quanti- 

 ties A, B, C, &c. The firft is. That it is not necefTary to com- 

 pute all of them feparately by feries, or by other methods : They 

 form a recurring feries; and the two firft being fo computed, all 

 the reft may be derived from them. The fecond thing is, That 

 the quantities A and B having been computed for any exponent 

 ;z, the correfpondent quantities are thence derived, by eafy for- 

 mulse, for the exponents //+i, «-4-2; n — i, n — 2j and in 

 general for the exponent n-\- m^m being any integer number, 

 pofitive or negative. 



From thefe remarks, it follows, that the whole difficulty lies 

 in the computation of the two firft quantities, A and B ; and 

 that we are not confined to a given exponent », but may 

 choofe any one in the feries, n-\-i, «-|-2, &c. ; « — i,« — 2,&c. ; 

 that will render the computation moft eafy and expeditious. 



ThuSl, in order to compute the quantities A and B, for the 



exponent , M. de la Grange makes choice of the expo- 

 nent + -, which, in the whole feries of exponents + |. + ^» 

 — \ — ^, &c. is the moft favourable for computation, on ac- 

 count of the convergency of the coefficients of the feries for A 

 and B. 



In 



