104 
[182 
182. 
ON LAGRANGE’S SOLUTION OF THE PROBLEM OF TWO 
FIXED CENTRES. 
[From the Quarterly Mathematical Journal, vol. ii. (1858), pp. 76—83.] 
The following variation of Lagrange’s Solution of the Problem of Two Fixed 
Centres( 1 ), is, I think, interesting, as showing more distinctly the connection between the 
differential equations and the integrals. The problem referred to is as follows: viz. 
to determine the motion of a particle acted upon by forces tending to two fixed 
centres, such that r, q being the distances of the particle from the two centres 
CL /3 
respectively, and a, /3, <y being constants, the forces are — + 2yr and + 2yq. 
Take the first centre as origin and the line joining the two centres as axis of x\ 
and let h be the distance between the two centres, then writing for symmetry 
x = x 1 = x 2 + h, 
(so that x x is the coordinate corresponding to the first centre as origin, and x 2 the 
coordinate corresponding to the second centre as origin) the distances are given by 
the equations 
r 1 = Xj 2 + y 2 + z 2 , q 2 = x 2 2 + y 2 + z 2 , 
and the equations of motion are 
d 2 x 
dt 2 
ax 1 /3x 2 
- 2 y0i + O, 
dt 2 r 3 q 3 
d 2 z az 7 z 
dt 2 r 3 q 3 ^ Z> 
1 Lagrange’s Solution was first published in the Anciens Mémoires de Turin, t. iv., [1766—69], and is 
reproduced in the Mécanique Analytique,