moving in Eccentric and Inclined Orbits, 3 



sions for the co-ordinates of w' in terms of the eccentric ano- 

 maly of m. Such expressions are very easy to obtain, and are 

 very convergent. It will be recollected that before I endea- 

 voured to develop the disturbing function in the lunar theory 

 in terms of the mean motions of the sun and moon, the inva- 

 riable practice had been (see MeQunique C6leste, vol. iii. p. 189) 

 to express the co-ordinates of the sum in terms of the true 

 longitude of the moon ; but the equation which connects the 

 eccentric anomalies of two bodies is far simpler than that which 

 connects the true anomalies, or xf and t;, and therefore the 

 conversion which I employ is made with greater facility. The 

 quantity under the radical sign in R may thus be considered 

 as a function, of which the general term can be represented by 



sin /. , ., n' \ 

 ^cosr + ^VV' 



a being a numerical quantity. The development of this 



13 

 quantity to the power — ^ or — -, may be facilitated by the 



use of tables, which give the numerical coefficients in the 

 development of {l—u4 cos a} ~^, {1— y4cos«}"% &c. Such 

 tables have been calculated for me by Mr. Farley. By pro- 

 ceeding in this way, no term is ever introduced which affects 

 the final result beyond a given place of decimals. For the 

 development of the radical admits of being exhibited in the 

 form 



>4 + -B+C+Z) + &c.; 

 such that 



5=a^(a, C=(35(a, D=:yC(a, 



so that each term is deducible from the one which precedes 

 it, by the multiplication of that term by a(B, /3®, &c., a, /3, 

 y, &c. being proper fractions. If therefore the terms in the 

 two quantities which form those products, such, for instance, 



as oiA and <^ 



which form B, are sorted and arranged in the order of their 

 numerical magnitude, as soon as any one partial product sinks 

 below any limit that may be assigned, all the succeeding terms 

 are necessarily of inferior magnitude ; and the approximation 

 stops, as it were, of itself, without any exercise of thought on 

 the part of the computer. 



When r > r', that is, when the planet disturbed is superior 

 to the disturbing planet, I am not able to suggest any other 

 course than to develop in terms of the true anomaly of the dis- 

 turbed planet, and the mean anomaly of the disturbing planet, 



B2 



