Partition of Kinetic Energy. 107 



Another feature worthy of attention is the spacial distri- 

 bution ; and it happens that this is peculiar in the present 

 problem. To investigate it we mu>t integrate (23) with 

 respect to u and. v, x and y being constant. Since x and y 

 are constant, V is constant ; so that, if w T e suppose E to lie 

 within narrow limits B and E + 6?E, the resultant velocity U 

 will lie between limits given by 



U<£U=dE (26) 



If we transform from ic, v to U, 0, where 



w = Ucos#, u = Usin0, . . . (27) 



dudv becomes XJdJJ d0; so that on integration with respect 

 to 6 we have, with use of (26), 



27rF(E)dE.dxdy (28) 



The spacial distribution is therefore uniform. 



In order to show the special character of the last result, it 

 may be well to refer briefly to the corresponding problem in 

 three dimensions, where the coordinates of a particle are 

 x, y, z and the component velocities are u, v, w. The steady 

 distribution corresponding to (23) is 



f(E)dxdydzdudvdw, .... (29) 

 in which 



E = iU 2 -fV=iu 2 + ^ 2 + iz^ + Y. . . (30) 



Here equation (26) still holds good, and the transformation 

 of du dv dw is, as is well known, 47rU 2 dTJ. Accordingly (29) 

 becomes 



4wF(E)tfE.(2E-2V)*<fa?<fy, . . (31) 



no longer uniform in space, since V is a function of a?, y. 



In (31) the density of distribution decreases as V increases. 

 For the corresponding problem in one dimension (18) gives 



F(E)dE.(2E-2V)-*d#, .... (32) 



so that in this case the density increases with increasing V. 



The uniform distribution of the two-dimensional problem 

 is thus peculiar. Although an immediate consequence of 

 Maxwell's equation (41), see (41) below, I failed to remark 

 it in the note before referred to, where I wrote as if a uniform 

 distribution in the billiard-table example required that Y = 0. 

 In order to guard against a misunderstanding it may be well 

 to say that the uniform distribution does not necessarily extend 

 over the whole plane. Wherever (E— Y) falls below zero 

 there is of course no distribution. 



