} 
; 
5 
513 
1845, the polar equation of the orbit, under the’ well-known 
form, 
(19) 
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, 
(m+ Am)du(fArdé) 
v(Ar§Ardé)  ’ 
(where m+ Am, + + Ar, are the mass and the vector of velocity 
of an attracting body, as m, + 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. 
: II. The vector function ¢(a) in (9) may be called the 
TRACTOR corresponding to the vector of position a, or simply 
r= ——_. 
1+ ecosv 
dr=> 
(20) 
the tractor of a; and another general transformation of this 
tractor, which is more intimately connected than the foregoing 
with the problem of perturbation, may be obtained by sup- 
posing the vector a to receive any small but finite increment 
3, 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 éractor 
¢({3 +a), (answering to the new vector 8+a, which is drawn 
from the beginning of a to the end of (3), according to the 
ascending powers of this added vector 3. In this manner we 
find 
¢(B+a)={—(B+a)}> (B+a) = 
{ —a%(1-4+a71B) (1+ Ba)} fal +1} 
= (14 Ba-")-*(1 +a"B)? a“"(— ce’); (21) 
o(B+a) = Zn w hn, ws (22). 
that is, 
