386 J. W. Gihhs — Equillhrhim of Heterogeneous Substances. 

 we may write 



This is true of reversible variations in Avhich the surfaces wliich have 

 been considered are fixed. It will be observed that i'' denotes the 

 excess of the energy of the actual mass which occupies the total 

 volume which we have considered over that energy which it would 

 have, if on each side of the surface S the density of energy had the 

 same uniform value quite up to that surface which it has at a sensi- 

 ble distance from it; and that ?/*, n4, m|, etc., have analogous signi- 

 fications. It will be convenient, and need not be a source of any 

 misconception, to call 6^ and if the energy and entropy of the surface 



t^ if 



(or the superficial energy and entropy), — and — the siqmficial den- 



lit TTt 



sities of energy and entropy, — i, — ?, etc., the superficial densities of 



the several components. 



Now these quantities (f'', 7/^, m\^ etc.) are determined partly by 

 the state of the physical system which we are considering, and partly 

 by the various imaginary surfaces by means of which these quanti- 

 ties have been defined. The position of these surfaces, it will be 

 remembered, has been regarded as fixed in the variation of the sys- 

 tem. It is evident, however, that tlie form of that portion of these 

 surfaces, which lies in the region of homogeneity on either side of the 

 surface of discontinuity cannot aftect the values of these quantities. 

 To obtain the complete value of dt^ for reversible variations, we have 

 therefore only to regard variations in the position and form of the 

 limited surface s, as this determines all of the surfaces in question 

 lying within the region of non-homogeneity. Let us first suppose 

 the form of s to remain unvaried and only its position in space to 

 vary, either by translation or rotation. No change in (492) will be 

 necessary to make it valid in this case. For the equation is valid if 

 !$ remains fixed and the material system is varied in position ; also, if 

 the material system and § are both varied in position, while their 

 relative position remains unchanged. Therefore, it will be valid if 

 the surface alone varies its position. 



But if the form of s be varied, we must add to the second member 

 (492) terms which shall I'epresent the value of 



6t^ — t 6}f — // , f5;/v^ — //„ dm\ — etc. 



due to such variation in the form of <*. If we suppose s to be suffi- 



