144 
PROF. H. F. BAKER ON CERTAIN LINEAR 
in which, as far as c 3 , 
X = 
and, as far as c 4 , 
v 7 2 
(l + ir c ')> 
U = 1 +c sin t + c 2 (i v 7 2 cos t — -g- cos 2 1) + c 3 ( f-g sin t + sin 
3i 
2A 
sin 3f i 
144 y 
+ c 4 ( ^ 1 i cos t —cos 2£ - 
11 • Q . , cos 4A 
-t= i cos 3 1 4-— . 
432\/2 4608/ 
§ 7. We pass now to the consideration of a pair of simultaneous differential 
equations arising in the consideration of the stability of the motion of three particles 
occupying the angular points of an equilateral triangle moving under their mutual 
gravitation. 
The stability of this motion has been discussed by Routh (‘ Proc. Lond. Math. 
Soc.,’ VI., 1875; ‘ Rigid Dynamics,’ II., p. 61) in the case when the relative paths 
of the particles are circles .* In what follows we do not assume this. 
-< 
The three particles being S, E, M, take an axis through S, say SX, rotating with 
angular velocity 6, the line SE being supposed to coincide very nearly with SX. 
Draw a perpendicular EH from E to SX, denote EH by y, and SH by A + x, where 
x, y will be considered small, their squares being neglected, but A is a variable finite 
quantity. Draw a second axes SY through S at a constant angle - with SX, and 
3 
* The following references may be of use: —Charlier, ‘Die Mechanik des Himmels,’ and ‘Astr. 
Nachr.,’ 193, 15; STOCKWELL, ‘ Astron. Journ.,’ 557 (1904); Linders,' ‘ Arkiv for Mat.’ (Stockholm), 
IV., No. 20; Brown, ‘Monthly Notices, R.A.S.,’ LXXI. (1911), pp. 439, 492; Heinrich, ‘Astr. 
Nachr.,’ 194, 12 (December, 1912); Block, ‘Arkiv for Mat.,’ X., 4 (1914). 
