230 The Evolution of Star -Clusters [en. x 



Within the small range dxdydz, the gravitational forces arising from the 

 universe as a whole will be sensibly constant ; suppose them derived from a 

 potential F. Then the motion of every star included in formula (553) must 

 be determined by the equations of motion, 



da_dV dv_dV dw_W 



dt'dx' dt~dy' dt~dz .................. (5 



After an interval dt, this group of stars must have velocity components 

 lying within a small range dudvdw surrounding the values 



dx dy dz 



while its position will be confined to a small region of space of extent dxdydz 

 surrounding the point 



x + udt, y + vdt, z + w dt. 



Hence, with the notation already introduced in formula (553), the number 

 of stars in this group must be 



f[u + ^dt, v + TT- dt, w -f- -= dt, x -f udt, y + vdt, z + wdt, t + dt ) 

 \ ox oy oz ) 



dudvdwdxdydz ...(555), 

 so that this expression must be equal to expression (553). 



Expanding expression (555) as far as first powers of dt, and equating to 

 expression (553), we obtain 



df ^ dV df , 9F df dV df df df df 



' _1_ v _|_ J . I / I n, J I n\ > I nit J __ Q / K K K. \ 



dt dx du dy dv dz dw dx dy dz 



This is the differential equation which must be satisfied by the distribu- 

 tion function y in every problem of stellar dynamics*. 



231. Being a linear equation in /, this equation may be solved in accord- 

 ance with Lagrange's rule. This rule directs us to find as many integrals as 

 possible of the system of equations 



7 du dv dw dx dy dz /ce>r\ 



* = 8? = 8? = 8F = == V = (557> 



dx dy dz 



If E l cons., E 2 = cons., . . . are all the integrals of these equations, then 

 the solution of the original equation (556) is simply 



/= <(#!, E 2 , ...) (558), 



* The student of the Kinetic Theory will recognise that it is simply Boltzmann's well-known 

 equation with the collisions left out. Cf. Boltzmann, Vorlesungen ilber Gastheorie, i. p. 132, 

 or Jeans, Dynamical Theory of Gases (2nd Ed.), p. 226. 



