345 , 346 ] Moving Clusters 373 
Any one star of a moving cluster will be acted on by forces of three kinds: 
(а) The forces arising from the general gravitational field of the galactic 
system as a whole. 
(б) The forces arising from other stars of the same cluster. 
(c) The forces from near stars, not belonging to the cluster, which are 
undergoing encounter with the star in question in the sense that the forces 
between the two stars are of appreciable amount. 
We shall only consider motion in the central dense parts of the galactic 
system, and here the star-density may be supposed to be approximately 
uniform, giving rise to a uniform mean density p of matter. If, following 
Kapteyn, the surfaces of equal density in the galactic system as a whole are 
supposed to be similar spheroids, then the gravitational forces in the central 
region of the galactic system will be derived from a potential 
V=K-A(x* + y*)-Cz* (346*1), 
where A, G,K are constants, the galactic plane being the plane of xy. The 
ratio of C to A depends only on the shape of the spheroids. With Kapteyn’s 
ratio 5*102 for their semi-axes, we find that (7 = 6*31A. Poisson’s relation 
V 2 F= — 4w7 p gives the further equation 2A + G= 2iryp, so that 
A = 0*755 7/ 3; C = 4 * 787,3 (346*2). 
Let the centre of gravity of the moving cluster be supposed to be at the 
point x, y, z. The components of force here are — 2 Ax, — 2 Ay, — 2Cz. At an 
adjacent point x + £, y + y, z+% the components of force are —2A (# + £), 
— 2A (y + y), — 2G(z+ £). Thus a star whose co-ordinates relative to the 
centre of the cluster are £, y, £, will experience an acceleration relative to the 
centre of the cluster, of components — 2 A%, — 2Ay, — 2 C£. Thus if forces (a) 
alone were operative, the equations of motion of a star in the cluster, relative 
to the cluster as a whole, would be 
a?—* 0 * < 346 ' 3 >- 
Since A and B are both positive, the resulting motion of the star in the 
cluster will be compounded of three harmonic oscillations, two parallel to the 
galactic plane, each of period 2n/(2A)^, and one perpendicular to this plane 
of period 2vr/(2(7)^. With an average of one star of mass 2 x 10 33 per ten 
cubic parsecs, the actual period of the oscillation parallel to the galaxy is 
found to be 236,000,000 years, and that of the oscillation perpendicular to the 
galaxy is 91,000,000 years. 
If we wish to include the forces ( b ) arising from the other cluster stars, we 
may assume the cluster to be of uniform density and of ellipsoidal shape. Its 
gravitational potential at a point whose co-ordinates relative to its centre are 
£, y, £, is then of the form