76 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



none of the assumptions which have been made, tacitly or otherwise, 

 relating to the formation of these new parts, shall be violated. These 

 assumptions are the following: that the relation between the varia- 

 tions of the energy, entropy, volume, etc., of any of the original parts 

 is not affected by the vicinity of the new parts ; and that the energy, 

 entropy, volume, etc., of the system in its varied state are correctly 

 represented by the sums of the energies, entropies, volumes, etc., of 

 the various parts (original and new), so far at least as any of these 

 quantities are determined or affected by the formation of the new 

 parts. We will suppose De, Dr\, Dv, Dm 1} Dm 2> . . . Dm n to be so 

 defined that these conditions shall not be violated. This may be 

 done in various ways. We may suppose that the position of the 

 surfaces separating the new and the original parts has been fixed in 

 any suitable way. This will determine the space and the matter 

 belonging to the parts separated. If this does not determine the 

 division of the entropy, we may suppose this determined in any 

 suitable arbitrary way. Thus we may suppose the total energy in and 

 about any new part to be so distributed that equation (12) as applied 

 to the original parts shall not be violated by the formation of the 

 new parts. Or, it may seem more simple to suppose that the 

 imaginary surface which divides any new part from the original is 

 so placed as to include all the matter which is affected by the 

 vicinity of the new formation, so that the part or parts which we 

 regard as original may be left homogeneous in the strictest sense, 

 including uniform densities of energy and entropy, up to the very 

 bounding surface. The homogeneity of the new parts is of no con- 

 sequence, as we have made no assumption in that respect. It may 

 be doubtful whether we can consider the new parts, as thus bounded, 

 to be infinitely small even in their earliest stages of development. But 

 if they are not infinitely small, the only way in which this can affect 

 the validity of our formulse will be that in virtue of the equations of 

 condition, i.e., in virtue of the evident necessities of the case, finite 

 variations of the energy, entropy, volume, etc., of the original parts 

 will be caused, to which it might seem that equation (12) would not 

 apply. But if the nature and state of the mass be not varied, equa- 

 tion (12) will hold true of finite differences. (This appears at once, 

 if we integrate the equation under the above limitation.) Hence, 

 the equation will hold true for finite differences, provided that the 

 nature and state of the mass be infinitely little varied. For the dif- 

 ferences may be considered as made up of two parts, of which the 

 first are for a constant nature and state of the mass, and the second 

 are infinitely small. We may therefore regard the new parts to be 

 bounded as supposed without prejudice to the validity of any of our 

 results. 



