776 Prof. J. H. Jeans on the 



and average according to the law (4). In this way we find 

 the average velocity components u', v', iv r after time t. The 

 vector i/, v r , w' must, from symmetry, be in the same direction 

 as u, v, iv : its amount will depend on u 2 + v 2 + iv 2 and on the 

 constants of the matter and the electrons. 

 Thus we must have 



v! = uf(r, it 2 + v 2 + ic 2 ), &c. ; 



and hence, on averaging for all values of u, v, ic, we find 

 that if it ' y v '. w f are the values of i/ , v , w after time t, 



u ' = u <j>(t, h) (5) 



If we could evaluate </>, this equation would be the gene- 

 ralization, in integral form, of equation (3) with X = 0. 



We can obtain some information as to the function <£. 

 We notice first that at time t = 0, 



du dy 



dt dx ' 



so that, on averaging, the value of -j~ is zero, and hence 



</>(t, 7i), for small values of t, is of the form 1 + r 2 /(/i). 



Next we consider the form of the function for large values 

 of t. Let t be so large compared with the time of encounter 

 with a molecule, that the velocities of those electrons which 

 originally had velocities u, v, w may be regarded as distributed 

 at random. Then, if u Q r, 9 v ", w n are the values of w , v , z% 

 after a time 2t, we shall have, in addition to equation (5), 



u " = u (J)(2t, It) ; « o " = w o '0(t, h), &c, 



whence it follows that 



and equation (6) can be put in the form 

 du .p 



provided dt is sufficiently large. This equation is exactly of 

 the form of (3) with X = 0. It follows that the general 

 equation, no matter what the nature of the motion of the 

 electrons, is of the type of equation (3). 



6. We now return to equation (3), which with a suitable 

 value for y will express the relation between X and u in any 

 motion in which the quantities do not change too rapidly 

 with the time. 



