2-p- 72\» ^ ^^ ^^^^ angle included by lines drawn from 



moving in Eccentric and Inclined Orbits. 5 



terms of which the coefficients are each below •! in numerical 

 value, and the quantity {1 +P}"i^ can be developed accord- 

 ing to powers of Q in a rapidly converging series. 



„ , 2)]^ a' r so 



1+P=1 ^-r— COS^+ljV ' 



(T ?•' a '^ 



2^_ 2aa' 



«'"2 r^ 

 the sun to m and m\ p= -^ -^ — 1. 



2y\^ a! r 

 The terms contained in r- cos 8 obviously exceed 



fj r a •' 



greatly in magnitude those contained in )j^^, 



- cos 8=2 cosy 4- k siny ; 



f being the true anomaly of m', and u the eccentric anomaly ofw, 

 i= — e^4-^cos 0— 'V^l —e^% sin y, 

 ^=— eC + C cosy + ^1— e^Ssiny; 



S[, 93, C and 19 being constants, each necessarily less than 

 unity, which depend only on the inclination of the orbits and 

 the position of their line of intersection, and such that when 

 they are in the same plane 



^ = ja and 33 = C. 

 The process is precisely the same in substance, whether the 

 orbit of m is highly eccentric and inclined, or circular, and in 

 the same plane with that of ;«' ; the only difference being that, 

 while in the former case it may be necessary to detach as many 

 as six terms to form the quantity 1+P+Q, in order that Q. 

 may not contain any term of which the numerical coefficient 

 exceeds •! in magnitude ; in the latter case, supposing e' the 

 eccentricity of w' to be inconsiderable, 1+P+Qwill only 



contain one factor, and therefore {1+P+Q}~^, &c. can be 

 calculated with greater facility. Thus, for example, in the 

 perturbations of 



Pallas by Saturn, it is convenient that 1+P+Q should 

 contain three terms. 



Pallas by Jupiter, it is convenient that 1+P+Q should 

 contain four terms. 



Encke's comet by Saturn, it is convenient that 1+P+Q 

 should contain five terms. 



Encke's comet by Jupiter, it is convenient that 1+P+Q 

 should contain six terms. 



