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



That part of the ay-momentum in the 5th cell which arises 

 from the motion of molecules with ^-velocities between u 

 and u + du, is 



7 W % K . QSTT 



b s mu = mu f- z, mu co rj Q sup 



n q =i n 



The total momentum in the sth cell, obtained by summa- 

 tion, is accordingly 



M 3 = = Tb s mu=Z£sm^, . . . (24) 

 where 



f 9 = X muarjq (25) 



11 = — 00 



Equation (24) analyses the total .v- momentum into the 

 momenta of trains of waves. The total kinetic energy of 

 these waves is 



i T^= I ^(?, 2 +?/+-C)- • • (26) 



s =iiNm 4JN?)i ' 



To find the law of distribution of the £"s we return to 

 equation (23). The law of distribution of mu o)7j u mu co t? 2 , . . . 

 may from this equation be expressed in the form 



A „ e -K>>{(mua, n tf+(mu<o n . 2 y+ ...} d{jnu ^ ^ d ( mu & %) ? 



where « ,/ =0 2 /4N / w 2 G> 2 m 2 . It follows at once that the law 

 of distribution of the quantities f l9 f 2 , ... given by equation 

 (25) is* 



A ,// «- e " ,( ^ ,+ V + -)d6d&..., . . . (27) 

 where 



1 «=+» 1 4o) 2 m 2 «™+- , , 4mNRT 



/C 1I-— CO «• AA 26= — QO /l 



* If m^, m^i, m^ are distributed according to the law 



_ A {(mitt 1 )2+(TOi«i)2+(«ii«; 1 )2J. 

 Ce mi d{m l u 1 )d(m 1 v 1 )d(m 1 w 1 ), 



and m 2 « 2 , . . • according to the law 



. |(«!ol t . : )2+...j. 



C e nl s * •> d(m 2 u 2 ) . . ., 



then m 1 w 1 + w?oMo+ . . •> • • • are distributed according to the law 



where M=«* 1 +m? 2 + '• •• • This result is obvious from physical con- 

 siderations; or may of course be obtained by algebraic transformation. 



