Molecular Velocities amongst the Molecules of a Fluid. 167 



and then left to itself, if we assume that there is no further 

 change of energy due to outside influences, then B in (1) is 

 constant. Hence 



8E = 0=§%(u n 8u n + v n 8v n + ic n 8iCn)mn- • • (3) 



Also, if we imagine small variations of the velocities that enter 

 into equations (2) to be made consistent with the kinematic 

 conditions of the fluid, we may write 



§8a.2,m n =§y i m n .8u n , (4) 



j*S& . 2m w =j2w w . Sv u , (5) 



^8c.^m n =^m n .8w n (6) 



We can now proceed to state the condition that the distri- 

 bution of the molecular velocities is the most probable. 



The probability that for a molecule taken at random the 

 component velocities are u, v, w, will be some function of 

 u, v, w, and may be expressed by F(«, v, w). 



The probability that a number of independent events may 

 all happen is the product of the separate probabilities. Hence 

 the probability that the component velocities of the molecules 

 composing the whole mass of fluid — considered as mutually 

 independent — may be respectively 



(«U Vl, Wl), (»«, «*, W*), Os, V 3 , W 3 ), 



is given by 



F(m x , V 1} tOj) F(« 2 , v 2 , w 2 ) ¥{u 3 , V 3 , IV 3 ) 



or, shortly, 



FxF.Fs (7) 



Expressing next the condition that the assumed distribution 

 of velocities is the most probable, we have, from (7), 



K£^ + f^ + £^)-w 



Dividing the terms by the continued product F x F 2 F 3 



we may write the result under the form 



o-K(£.n + £..* + £^). • • («) 



We have therefore five equations, viz. (3), (4), (5), (6), and 



N2 



