234 Prof. J. H. Jeans on Temperature-Radiation 



New functions u l9 u 2 , . . . u n and L' of these variables are 

 introduced, defined by 



Wl= ^-r-, &C 



L' = u-fix + i( 2 2 + + u n 6n— L ; 



-oved in the usual way, by 

 , that 



du x _ BIZ. de Y _~bV % 



and it is then proved in the usual way, by purely algebraic 

 transformation *, that 



(5) 



dt ~d6 x dt "diii 9 



whence it follows that 



^_(duA + 1«\ =0 r6) 



'du 1 \dt)^-d6 1 \dt) W 



The condition that ^~Ldt is to be stationary determines 

 uniquely the changes in ± , 6 2 , . . . nj #1, #2> • • • Q n -> starting 

 from given initial values. Now let a generalized space be 

 constructed, having h 2 , • • • @n, u 1} u 2 . . . w«, as its coor- 

 dinates. Let this space be filled with " representative points " 

 each of which is to move as directed by the condition that 

 \~Ldt is to be stationar}^. If p denote the density of these 

 u representative points " at any point in the generalized space, 



and if j£ denotes the rate of increase of p as we follow the 



" representative points " in their motion, we have f, as a 

 matter of algebraic calculation, 



Dp 

 Dt 



by equation (6). If the "representative points" tended, in 

 their motion, to concentrate onto any special points or regions 



in the generalized space, then ~ would be positive for those 



points or regions : similarly, if the " representative points " 



tended to scatter, j~- would be negative. The result obtained 



in equation (7), that j~ vanishes everywhere, shows that 



there is no tendency for the a representative points " to 



* Houth, 'Elem. Rigid Dynamics/ chap, viii., or Jeans, 'Theoretical 

 Mechanics/ chap. xii. 



f Jeans, ' Dynamical Theory of Gases/ p. 63. 



