ENERGY ACCELERATIONS, A STUDY IN ENERGY, &C. 291 



(2) The mean values [wr] and [y?^] are not equal unlessthefollowing 

 condition is satisfied: 



[X2] [ff - [Y''] [rY + [X2 _ J.2] I ^,j ^^j _ a «2 j 



+ ;>[jr]|w--[/]|H = o. 



This is satisfied in particular by 



[X2] = [r2] and [r-] = [tl 



From thèse two results we may say that [w^] = [y-] and [nv'] = 

 if the probable field of force be isotropic but not otherwise except when 

 certain complicated conditions are satisfied. The ordinary transforma- 

 tion formulae however show that there is always one pair of axes for 

 which [u i'] = and [?^"] and [y-] are a maximum and minimum res- 

 pect! vely. 



(3) The fourth déterminant in équations (5) reduces to 



This vanishes if [r -f- /] ^ 0, that is if the mean value of d'^ V\ch^ 

 -\- d-Vjdi/- = 0. It follows that if the field of force is known to satisfy 

 La place' s Equation, we shall hâve a failing case in which the distribu- 

 tion of squares and products of velocities may be expected to assume 

 an altogether exceptional character. 



Eveu in the corresponding problem in three dimensions, the pro- 

 perty that the Newtoniau potential satisfies Laplace's équation in free 

 space cannot fail to produce some modifications in the équations of 

 energy equilibrium when the law of force is that of the inverse square, 

 but whether such modifications hâve any physical interprétation in 

 cosmic phenomena must be regarded, at any rate for the présent, as a 

 matter of pure spéculation. 



I hâve formed the déterminant for the three dimensional problem cor- 

 responding to that just considered but it does not appear that the mean 

 value of V' V is in this case a factor of the déterminant. 



Sutmnarij . — (1). Clonsidering the matter as a problem in pure dyna- 



19* 



