371 
342-345] 
Star-Streaming 
During this process the form of the function / must change, and when 
the final steady state is attained, the general principles of statistical mechanics 
indicate * that the form of the function f must be 
f(E u *r 3 ) = Ae- 2hm(El + u,m3) (343-2) 
where A, h and co are constants. Inserting their values for E 1 and nr z , the 
law of distribution becomes 
f(E nr) — Ae~ ^ m i( u2 + v2 + wS ) ~ 2J / +2w (xv-yu)] 
_ ^g~hm[(u-wy) 2 +(v + u>x) 2 + w 2 ] + 2h[V+}u>' 2 (x 2 + y 2 )] ( 343 - 3 ) 
Integrating over all values of u, v, w from — 00 to + 00 , we find that the 
density at x, y, z must be of the form 
p==C e 2hiV+W{x2+y2)] (343-4) 
where G is a constant. Thus, just as in a rotating mass of gas, the surfaces 
of equal density have equations of the form 
V+^co 2 (a? + y 2 ) = cons (343*5). 
These surfaces have already been discussed theoretically in Chapters vn 
and viii. 
344. Formula (343*3) shews that, when the steady state is attained, the 
stars at any point have a mass-velocity of components - toy, cox, 0, which is 
the velocity of a pure rotation with angular velocity co about the axis of z. 
Superposed on to this mass-velocity are velocities of individual stars distributed 
according to a Maxwellian distribution. In this motion there is no star 
streaming. 
Thus the fact of star-streaming being observed is evidence that the stars 
are not yet in the final steady state now under consideration. As we have 
seen, star-streaming can be explained on the supposition that they are in a 
state of steady motion which is permanent except for the effects of near 
encounters, but they must still be far from the later state in which even near 
encounters do not disturb their motion. The discussions of § 272 and § 276 
have nevertheless suggested that they may be a good distance on towards 
this final state. 
345. The law of density (343*4) which must obtain in the final state giv^s 
infinite density at an infinite distance from the centre except when co = 0 . 
Even when co = 0, it gives a finite density at all distances from the centre, so 
that the system of stars is of infinite extent in space; it is in fact arranged 
like a mass of gas in isothermal equilibrium without rotation. 
When co is different from zero, the formula shews that there can be no 
steady state until all the stars have been scattered to infinity. Actually, as 
we have seen (§ 229), the surfaces of equal density (343*5) consist of some 
closed surfaces and some open surfaces. If the density at the last of the 
* Dynamical Theory of Gases (4th ed.), §§ 107, 113.