142 BUTLER 



ART. D 



having been formed by an infinitesimal variation in the state or 

 composition of a part of the original mass. The new parts 

 form.ed in an infinitesimal variation of the original mass are 

 necessarily infinitely small. Let De, D-q, Dv, Drrii,. . .Drrin 

 denote the energy, entropy, volume and the quantities of 

 the components 8\, . . .Sn contained in any one of these new 

 parts. We have no right to assume that a very small 

 new part is homogeneous or that it has a definite physical 

 boundary. Under these circumstances in order that these 

 quantities may have a definite meaning it is necessary to define 

 unambiguously the boundaries of the new parts. Gibbs uses a 

 convention similar to that which he employs in the theory of 

 capillarity. A dividing surface is drawn round each new part in 

 such a way that it includes all the matter which is affected by the 

 vicinity of the new part, so that the original part or parts remain 

 strictly homogeneous right up to this boundary surface. De, 

 Dtj, Dv, etc., then refer to the whole of the energy, entropy, 

 volume, etc., within the boundary surface. 



If we use, as before, the character 5 to express infinitesimal 

 variations of the original parts of the system, the general con- 

 dition of equilibrium may be written in the form 



(25e + 2Z)e), ^0 (186) [36] 



or, substituting the value of SSe taken from equation (62), 



SDe + 2^577 - 'L'pbv + ^/xiSmi . . . + SM«5wn ^ 0. (187) [37] 



Making use of this equation Gibbs deduces de novo and by a 

 very general argument the conditions of equilibrium when the 

 component substances are related by r equations of the type: 



ai ©1 + a2 ©2 ... + a„ ®„ = 0. (188) [38] 



We shall consider here the simpler case in which the components 

 ^1, Si,. . . Sn are all independent of each other. There is no 

 real loss of generality in this limitation for, as Gibbs points out, 

 we may consider all the bodies originally present in the system 

 and the new bodies which may be formed to be composed of the 

 same ultimate components. 



