208 SCIENCE PROGRESS. 



for the first, and 



Q' x and Q' x + <tQ\ \ 



Q' a and Q' a + dQ' 2 | A' 



etc., ; 



and 



P' x an d P\ + rtF, n 



P' 2 and P' 2 + dY 2 I B' 



etc., ) 



for the second molecule. Let this be the state when 



mutual action ceases. 



Let fdq\ . . . dp '„ be the number per unit of volume 

 of molecules in the state a b' , and F'dO' l . . . dV „ the 

 number in the state A' B'. Then shortly f'Y'dq\ . . . 

 dP' '„ is the number of pairs in the second state. 



It is now required that the number of pairs in the first 

 state shall be the same as in the second, that is./Fdq, . . . 

 dP n => f'Y'dq\ . . . dV n . This has been shown by Boltz- 

 mann to be necessary for permanence. But since q\ . . . P'„ 

 are functions of q 1 . . . P„ and of the time t and no other 

 quantities whatever, dq 1 . . . dP n = dq\ . . . dP'„. There- 

 fore our requirement is satisfied if/T =/'¥'. 



n. If this equation be given only for particular values 

 of the variables q l . . . P'„, we cannot draw from it any 

 inference concerning the form of the functions f and F. 

 But if it be satisfied for all possible initial values of the 

 variables q l . . . P„ with the final values q\ . . . P'„ into 

 which they pass under the influence of their mutual action 

 alone, we can infer that the form of the function is/"= F = 



12. The theorem, as given by Watson, thus de- 

 pends on three conditions: (i) There must be an infinite 

 number of pairs, including all possible combinations of the 

 variables ; and (2) each pair must be uninfluenced by any 

 third molecule during the time of action of the mutual forces 

 between its members. (3) The fact of the near approach 

 of the two molecules to one another, which is implied in 

 their encounter, must not of itself destroy the independence 

 of the chances/ and F. This condition involves the pro- 

 perty mentioned above that the molecules must be " sparsely 

 scattered ". For, if they be crowded, this independence 



