68 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



will remain unchanged. Now as all values of the variations which 

 satisfy equations (24) will also satisfy equations (25), it is evident 

 that all the particular conditions of equilibrium which we have 

 already deduced, (19), (20), (22), are necessary in this case also. 

 When these are satisfied, the general condition (23) reduces to 

 M 8m l + 1T 2 2 <$ w 2 + Jf 8 2 Sm s ^ 0. (27) 



For, although it may be that ///, for example, is greater than M v 

 yet it can only be so when the following Sm^ is incapable of a nega- 

 tive value. Hence, if (27) is satisfied, (23) must also be. Again, if 

 (23) is satisfied, (27) must also be satisfied, so long as the variation 

 of the quantity of every substance has the value in all the parts of 

 which it is not an actual component. But as this limitation does not 

 affect the range of the possible values of 2m 1 , S$m 2 , and Em 3 , 

 it may be disregarded. Therefore the conditions (23) and (27) are 

 entirely equivalent, when (19), (20), (22) are satisfied. Now, by 

 means of the equations of condition (25), we may eliminate 'ZSm l 

 and 2$w 2 from (27), which becomes 



- a Af X 2 Sm 3 - b M< Sm 3 + (a + b) M< 8m 3 ^ 0, (28) 



i.e., as the value of 2 <5m 3 may be either positive or negative, 



aM l + b M z = (a + 6) M (29) 



which is the additional condition of equilibrium which is necessary 

 in this case. 



The relations between the component substances may be less 

 simple than in this case, but in any case they will only affect the 

 equations of condition, and these may always be found without .diffi- 

 culty, and will enable us to eliminate from the general condition of 

 equilibrium as many variations as there are equations of condition, 

 after which the coefficients of the remaining variations may be set 

 equal to zero, except the coefficients of variations which are incapable 

 of negative values, which coefficients must be equal to or greater 

 than zero. It will be easy to perform these operations in each par- 

 ticular case, but it may be interesting to see the form of the resultant 

 equations in general. 



We will suppose that the various homogeneous parts are considered 

 as having in all n components, 8 V $ 2 , . . . S n> and that there is no 

 restriction upon their freedom of motion and combination. But we 

 will so far limit the generality of the problem as to suppose that 

 each of these components is an actual component of some part of 

 the given mass.* If some of these components can be formed out 



*When we come to seek the conditions of equilibrium relating to the formation of 

 masses unlike any previously existing, we shall take up de novo the whole problem 

 of the equilibrium of heterogeneous masses enclosed in a non-conducting envelop, 

 and give it a more general treatment, which will be free from this limitation. 



