524 
PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
1. Let M represent the mass of the principal body, or its attraction at the unit of 
distance, m that of the body whose motion it is proposed to investigate, and ni! that 
of the disturbing body. The principal body is supposed to be at rest, and the rectan- 
gular coordinates and distances of the other two reckoned from its centre as origin, 
are respectively x, y, z, r and x', y\ z', r', at the time t reckoned from a given epoch. 
Then if [ju be put for M+m, and R for the expression 
m' [pox' -f yxj + zz') 
{[x'-xf+{y'-'!/)^ + {z'-z)^y’ 
we have for determining the motion of m the known equations, 
d'^x y,x 
dt'^' f.3 ~r 
(1.) 
df- ' r^' dy 
(2.) 
d^z y.z 6?R 
df^ ' dz ’ ■ 
(3.) 
Analogous equations apply to the motion of m' as disturbed by m. The directions of 
the axes of coordinates are entirely arbitrary. Conceive, therefore, to be known at 
a given instant (To) the position of the plane passing through m in the direction of 
its motion at that instant, and through the centre of M, and let this plane be the 
plane of xy. Conceive also to be known at a given instant (To) the position of the 
plane passing through m' in the direction of its motion at that instant, and through 
the centre of M, and let this plane make with the other the angle Also let the 
intersection of the two planes be the axis of x 
2. Now since according to these suppositions, the body m would continue in the 
plane xy if the disturbing force of m' ceased at the time To, it is clear that the coordi- 
nate z at any time To-j-r is a small quantity of the order of the disturbing force. By 
multiplying the equations (1.), (2.), (3.) respectively by 2dx, 2dy, and 2dz, adding, 
and integrating, we obtain 
( 4 , 
being the differential coefficient of R with respect to x, y, and ^ considered as 
rf(R) 
dt 
dz^ d dz 
functions of the time. But from what is said above, and — — ^ are of the order 
dr dz dt 
of the square of the disturbing force. Hence as it is proposed to conduct the 
approximation according to the powers of the disturbing force, these teians in the 
first and second approximations must be omitted. Also, if 6 be the polar coordinate 
Z^ 
of m reckoned on the primitive plane of its orbit from the axis of x, and ^ &c. be 
neglected, x=r cos 6,y=:r sin 0, r being now regarded as the projection of the distance 
on the plane of xy. Consequently 
