EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 73 



obtain conditions in regard to T, P, M lt M 2 ..M n , some of which 

 will be inconsistent with others or with equations (43). These con- 

 ditions we will represent by 



4^0, ^0, etc, (46) 



A, B, etc. being linear functions of T, P, M v M 2 , ... M n . Then it will 

 be possible to deduce from these conditions a single condition of the 



form 



a4 + /3+etc.^O, (47) 



a, /8, etc. being positive constants, which cannot hold true consistently 

 with equations (43). But it is evident from the form of (47) that, 

 like any of the conditions (46), it could have been obtained directly 

 from (45) by applying this formula to a certain change in the system 

 (perhaps not restricted by the equations of condition (39), (40), (41)). 

 Now as (47) cannot hold true consistently with eqs. (43), it is evident, 

 in the first place, that it cannot contain T or P, therefore in the 

 change in the system just mentioned (for which (45) reduces to (47))- 



2<ty + 21ty = 0, and 2&; + 2Dt; = 0, 



so that the equations of condition (39) and (40) are satisfied. Again, 

 for the same reason, the homogeneous function of the first degree of 

 M I} M 2 , . . . M n in (47) must be one of which the value is fixed by 

 eqs. (43). But the value thus fixed can only be zero, as is evident 

 from the form of these equations. Therefore 



(48) 



for any values of M v M 2 , . . . M n which satisfy eqs. (43), and therefore 



(49) 



for any numerical values of <S P @ 2 , . . . @ n which satisfy eqs. (38). 

 This equation (49) will therefore hold true, if for r of the letters 

 @ lf @ 2 , . . . <S n we substitute their values in terms of the others taken 

 from eqs. (38), and therefore it will hold true when we use j, 

 @ 2 @ n as before, to denote the units of the various components. 

 Thus understood, the equation expresses that the values of the 

 quantities in the parentheses are such as are consistent with the 

 equations of condition (41). The change in the system, therefore, 

 which we are considering, is not one which violates any of the 

 equations of condition, and as (45) does not hold true for this change, 

 and for all values of T, P, M v M 2 , . . . M n which are consistent with 

 eqs. (43), the state of the system cannot be one of equilibrium. 

 Therefore it is necessary, and it is evidently sufficient for equilibrium, 

 that it shall be possible to assign to T, P, M lf M 2 , ... M n such values, 



