513 



1845, the polar equation of the orbit, under the- well-known 

 form, 



r = -rr- • ('9) 



1 + ecosv 



This sketch of a process for employing the general transfor- 

 mation (10) in the theory of a binary system, may make it 

 easier, than it would otherwise be, to understand how the fol- 

 lowing equation for the motion of a multiple system, 



^'-^ v(Ar.SArdO ' ^ ^ 



(where m + Am, t + Ar, are the mass and the vector of velocity 

 of an attracting body, as m, r are those of an attracted one, 

 which is analogous to, and includes, the equation (13) for the 

 motion of a binary one, and which agrees with a formula com- 

 municated to the Academy in December, 1846), was obtained 

 by the present author ; and how it may hereafter be applied. 

 III. The vector function ^(a) in (9) may be called the 

 TRACTOR corresponding to the vector of position a, or simply 

 the tractor of a ; and another general transformation of this 

 tractor, which is more intimately connected than the foregoing 

 with the problem oi perturbation, may be obtained by sup- 

 posing the vector a to receive any small but finite increment 

 j3, representing a new but shorter vector, which begins, or is 

 conceived to be drawn, in any arbitrary direction, from the 

 point of space where the vector a ends; and, by then developing, 

 in conformity with the rules of quaternions, the new tractor 

 ^(fi + a), (answering to the new vector j3 + a, which is drawn 

 from the beginning of a to the end of j3), according to the 

 ascending powers of this added vector j3. In this manner we 

 find 



0(/3 + a)=:{-(i3 + «)n-*O + a)-' = 



{ -«Xl+a-'/3)(l+/3a-')}-Ha(l +cr'ft)}-' 



- (I+|3a-yHH-a-'/3)-ia-'(-«')^; (21) 



that is, 



(pQ3 + a) — ^n,n' fj)n,n', (22)- 



