Stellar Encounters 
321 
287-289] 
must increase as the mass of the encircled matter decreases, the relation 
Ma = constant found in § 268. If, as is more probable, the galaxy must be 
treated as a system of independently-moving stars, Ma must still remain 
constant, where a is the radius of the orbit of a single star surrounding the 
galaxy. Whichever way we regard the matter, the radius of the galaxy must 
be expanding approximately in accordance with the relation Ma — constant. 
Thus when, if ever, the galaxy had four times its present mass, its radius 
must have been only a quarter of its present radius, the star density must 
have been 64 times as great as now, and the density of matter 256 times 
as great as now. Formula (287'5) shews that if stellar velocities were the 
same as now, progress towards equipartition would have proceeded at 1024 
times the present rate, and division by a factor of 1024 reduces the time- 
interval just calculated from 5 x 10 13 years to 5 x 10 10 years. 
289. We are probably not entitled to assume that in these earlier epochs 
stellar velocities were the same as now, for the investigation of § 269 has 
suggested that loss of mass by radiation must in general be accompanied by a 
slowing down of stellar motion. 
Poincare’s Theorem (§ 62) shews that, when a system of stars is in steady 
motion, the mean square of their velocity is proportional to the mean gravi 
tational potential throughout the mass, and therefore approximately to M/a. 
If a varies inversely as M, the average stellar velocity must be proportional 
to the average stellar mass, a general result of which § 269 has already pro 
vided an illustration. Thus if the galaxy was then, and is now, in a state of 
steady motion, the velocities when the masses were four times as great as now 
must have been four times as great as the present velocities. 
Let M denote the present mass of a single star which we take to be 
typical of the galactic system as a whole. From formula (118‘4) we may 
suppose that at a time t back, its mass M' was given by 
where a = 5'2x 10 -88 . If, as we have supposed, V 0 varies as M', while v varies 
as M’ 3 , equation (287’4) shews that the rate of change of T' 2 varies approxi 
mately as M' 3 , the logarithmic function varying by so little that its changes 
may be disregarded. 
Thus if C denotes the present rate of change of ‘4 r2 , the rate of change at 
a time t back is given by 
i (>F 2 ) = C 
dt K ' 1 — 2aii/*’ 
and the change S'P 2 produced in T' 2 in the whole time t is 
between the radius a and the mass M of the galaxy being the relation 
8'P 2 = 
0 
1 - 2atM* dt