i82 RECTIFICATION of the ELLIPSIS, &c. 



feries of this form, A+ B cofip + C cof2(p + &c. The quan- 

 tities a and b reprefent the diftances of the difturbing planets 

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

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



Earth), that tlie fradlion - 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 qi;adratures, 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 neceffary 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 

 », the correfpondent quantities are thence derived, by eafy for- 

 mulae, for the exponents n+i, «-f2; « — i, « — 2; 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 ff, but may 

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

 that will render the computation moft eafy and expeditious. 



Thusi, 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 "f f • + ^ 

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



cov^nt of the convergency of the coefficients of the feries for A 



and B. 



In 



