232 Tlie Evolution of Star-Clusters [OH. x 



We may however put 



II lfdudvdw = v, 



1 1 \fuvdudvdw vwv, etc., 



where wv denotes the mean value of uv for stars of the first class at the point 

 x, y, z. Thus equation (563) becomes 



^(^u)+^( V u z )+~( v lw) + ^( v mu) = v d ^ ......... (564). 



dt^ dx dy^ dz^ dx 



When there is only one type of star, this and its two companion equations 

 are simply the hydrodynamical equations of motion of the element dxdydz of 

 the star-cluster. They could be derived directly from the equations of motion 

 (557) by the methods of the Theory of Gases*. When there are several types 

 of star we merely multiply the equations such as (564) by M, M f , etc. and 

 add, and the resulting equations are then seen to be the hydrodynamical 

 equations of motion of the element dxdydz. 



Thus it appears that if equations such as (558) are satisfied by/,/ 7 , etc., 

 then the hydrodynamical equations will be satisfied at every point of the star- 

 cluster in addition to the equation of continuity being satisfied. Every such 

 solution will accordingly give a possible motion of the stars of a cluster in a 

 field of potential V. In order that the field may be a purely gravitational 

 field, V must further be such as to satisfy equation (562), while we must still 

 further have V T if the stars move purely under their mutual gravitational 

 forces. 



Steady Motion 



234. The simplest problems of stellar dynamics naturally occur when the 

 group of stars under consideration is supposed to be in a steady state. The 

 steady state problem is the analogue of determining the configurations of 

 equilibrium for a gravitating mass of gas and we shall at once find that there 

 is a considerable similarity between the two solutions. 



Analytically the characteristic of a steady state solution is that /must be 

 independent of the time; the integrals E lt E 2 , ... which enter into / must 

 therefore not involve the time. Equations (557) reduce to 



du dv dw dx dy dz 



TiT = rTr = TTr = = = - - .................. (56o), 



dV 3V dy u v w 

 dx dy dz 



and one integral can be written down at once, namely the equation of energy 



Ei=% (u* + v 2 + w 2 ) - V = constant. 

 Jeans, Dynamical Theory of Gases (2nd Ed.), Equations (454), p. 180. 



