366 
The Galactic System of Stars [ch.xiy 
directions in space, for this would require that / should depend on c only. 
The velocity-diagram for the motions of the stars near a given point will not 
be spherically symmetrical, but will be a figure of revolution, having the 
radius through the point to the centre of the system as origin. 
H. H. Turner at one time suggested* that the observed star-streaming 
in our own universe might be explained in this manner, supposing it to 
orginate in the backward and forward motions of stars describing orbits 
of high eccentricity about the centre of the whole system. Eddington sub 
sequently shewed f that steady states of the type included in formula (335-4) 
were possible, but failed to notice that (as we shall soon see) they were only 
possible in a strictly spherical universe. It is fairly certain that our system 
of stars is nothing like spherical, being a lenticular or biscuit-shaped structure, 
and the observed star-streaming is almost certainly not along radii but 
nearly at right angles to radii. Thus a formula of the type of (335*4) cannot 
account for the observed stellar motions in the galactic system. 
Finally, we may notice that the total angular momentum of a system of 
stars whose steady motion is determined by this formula is zero, so that this 
state cannot be assumed by a system of stars which has originated out of a 
rotating nebula or other rotating body. 
Systems with Axial Symmetry. 
336. After the system whose gravitational field is spherically symmetrical, 
the next system in order of simplicity is one in which there is symmetry 
about an axis, so that the surfaces, both of equal potential and of equal 
density, are surfaces of revolution 
Taking the axis of symmetry to be the axis of z, the only integrals of 
equations (334'1) are seen to be the energy-integral, and the integral 
sr 3 = cons., which expresses that the moment of momentum of a star about 
the z-axis remains constant. Hence the only possible state of steady motion 
is one in which the law of distribution is of the form 
f(E u O (336-1). 
Systems with no symmetry. 
337. The only remaining type of system is that in which there is no 
symmetry at all. Here the only integral is the energy integral, so that the 
law of distribution must be of the form 
f(E x ) s/[i(tt* + v*+tt>»)-F] (337*1). 
Integrating over all values of u, v and w, we find that the density at any 
point is a function of V only, so that there must be a relation of the type 
V=<f>(p). 
* M.N. lxxii. (1912), pp. 387 and 474. 
t Ibid, lxxiv. (1914), p. 5; lxxv. (1915), p. 366; and lxxyi. (1916), p. 37.