335-339] Dynamical Discussion 367 
Now this is precisely the type of relation we obtained in Chapter viii 
in discussing the configurations of equilibrium of compressible masses. But 
when such a system is acted on by no forces except its own gravitation, the 
only solution of the equation represents a spherically symmetrical con 
figuration in which both V and p depend only on r. This brings us back 
to the spherically symmetrical configurations discussed in § 335, of which 
the systems now under discussion are seen to form a special class. Thus the 
search for systems in steady motion with no symmetry at all has failed; no 
such motion is possible. 
Stability. 
338. We are left with the result that, apart from spherical systems in 
which the spherical symmetry is complete, the only possible states of steady 
motion are those included in formula (3361), which represents motions and 
configurations possessing symmetry about an axis. The total angular momenta 
of these systems is not necessarily zero, and all systems formed out of rotating 
masses must be of this type. 
Formula (336T) includes the spherically symmetrical systems discussed in 
§337 as a particular case. It does not include the spherical systems with 
radial star-streaming discussed in § 335, and the mere fact that it does not at 
once suggests that these latter systems are of a special artificial kind. They 
have no angular momentum themselves and possess no counterparts which 
have angular momentum. Thus the slightest amount of angular momentum 
imparted to such a system destroys the state of steady motion. It must then 
pass through a series of states of unsteady motion until it ends up in one of 
the states specified by the law f(E 1} ct 3 ). These spherical systems with radial 
star-streaming must accordingly be regarded as unstable, and the only states of 
stable steady motion are included under the law f(E 1 , ct 3 ); their configurations 
possess symmetry about one axis, which may be thought of as a sort of axis 
of rotation. 
Stable States of Steady Motion. 
339. To consider these states of steady motion more fully, let us transform 
to cylindrical co-ordinates ■cr, 6, z and let the components of the velocity c in 
these three directions be denoted by II, 0, Z. Then w 3 = w@, and the law of 
distribution f(E 1) ot 3 ) assumes the form 
f[%(W+®* + Z 2 )-V,KT 0] (3391). 
The velocities are not distributed uniformly in space but shew preferential 
motion in the directions ± 0, i.e. in directions parallel to the central plane 
z = 0, and at right angles to the radius to the centre of the system. 
Thus the observed star-streaming in the galactic system exhibits the 
qualities predicted by formula (339*1) and, so far as these qualities go, the 
star-streaming is capable of interpretation as resulting from steady motion of 
the only kind which is dynamically stable.