333-335] Dynamical Discussion 365 
Clearly, however, equations (334T) are merely the equations of motion 
of the star in the general gravitational field of the whole system, so that 
E 1} jB'a, ... are the first integrals of the equations of motion. 
One such integral can be written down at once, namely the integral of 
energy 
E 1 = \ ( u 2 + v 2 + w 2 ) — V— constant (334’3), 
and it is only in special cases that other integrals exist. 
Systems with Spherical Symmetry. 
335. If the gravitational field of the cluster is spherically symmetrical, 
so that V is a function only of r, the distance from the centre, equations 
(3341) assume the form 
du _ dv _ dw _dx _dy _dz (‘■lajv'n 
xdV ydV z dV u v w 
r dr r dr r dr 
and there are three integrals 
ra-j = yw — zv = cons., 
OT 2 = zu — X w = cons., 
■GTj = XV — yu = cons., 
expressing that the moments of momentum -arj, «r 2 , -bt 3 per unit mass about 
the axes of co-ordinates remain constant. 
There are no other integrals except for special values of V. For instance, 
if V — a r 2 , where a is constant, there are additional integrals of the type 
u 2 — 2ax 2 = cons., etc., the corresponding motion being one in which each 
particle describes a continually repeated elliptic orbit about the centre. 
Apart from special artificial cases such as this, there can be no integrals 
beyond those already mentioned, so that the distribution-function f must be 
of the form 
f(E 1 , ■ur 1> ot 2 , ®- 3 ) dudvdwdxdydz (335'2). 
Since V 2 F= —47rp, it follows that if the gravitational field is spherically 
symmetrical, p can depend only on r, so that the density is arranged in 
spherical shells. The density is obtained by integrating the distribution 
law (335‘2) with respect to all values of u, v and w. For the resulting 
density to depend only on r, it is necessary (as is most easily shewn by 
rotation of axes) that this law should be of the form 
f(E lt ■ot 1 2 + -sr 2 2 + t!r 3 2 )dudvdwdxdydz (335 - 3). 
If c 2 is written for u 2 + v 2 + w 2 , the law of distribution can be written in 
the form 
V, r 2 (f sin 2 a) (335‘4), 
where a is the angle between the directions of r and c. At any single point 
in space the law of distribution f depends on c and on a. With this law of 
distribution, the velocities of the stars are not uniformly distributed over all