364 
The Galactic System of Stars 
[ch. XIV 
kinetic theory of gases with the collisions left out and a gravitational field 
thrown in. Just as, in the kinetic theory, the gas may be imagined divided up 
into a system of showers of parallel-moving molecules, so in stellar dynamics 
the stars may be imagined divided up into a system of parallel-moving clusters. 
But there is the essential difference that in stellar dynamics these clusters 
retain their identity through long periods of time, whereas in gas-theory they 
do not. 
Confining our attention to a small region dxdydz of space, let us suppose 
that the number of stars within this region, whose velocity-components u,v,w 
lie within prescribed limits dudvdw, is 
f(u, v, w, x, y, z, t) dudvdw dxdydz (333'1). 
If V is the gravitational potential of the whole system of stars, the motion 
of each of these stars will be determined by the equations 
du^dV dv = dV dw_dV 
dt dx ’ dt dy ’ dt dz 
After a time dt the parallel motion of these stars will take them to 
a position x + udt, y + vdt, w + zdt, while their gravitational accelerations 
dV 
will have increased their velocity components to w + r- dt, etc. Hence 
ox 
expression (333’1) must be equal to 
/ dV dV dV \ 
f\ +1 bx V d~ ^ w + dt, x + udt, y + vdt, z + wdt, t + dtj , 
since the stars specified in both groups are identical. We must accordingly 
have 
df dVdfdVdf dV df df df df „ 
+ a + t~ f- + uf-+vf + wf = 0... (333-2). 
dt ox ou dy ov dz ow ox oy oz 7 
This is the differential equation which must be satisfied by the distri 
bution-function f throughout any motion whatever of a system of stars. It 
will be seen to be identical with the corresponding equation in the kinetic 
theory of gases, except that the terms arising from collisions are left out. 
334. If the stars are in a state of steady motion, f does not change with 
the time, so that the first term d// dt in equation (333‘2) must be omitted. 
To solve the resulting equation for f Lagrange’s rule directs us to find all 
possible integrals E x = cons., E 2 = cons., etc., of the system of equations 
du dv dw dx dy dz 
sV = dV = dv=T = 'J = w (3m >- 
dx dy dz 
The solution of the equation is then simply 
/=<¿>0^, E 2} ...) (334-2) 
where </> is any arbitrary function.