J. W. Gibbs — Equilibriuni of Heterogeneous Substances. 497 



will not in this case be entirely separable from those relating to the 

 adjacent surfaces, since a displacement of the line will involve a dis- 

 placement of the whole side of the crystal which is terminated at this 

 line. But the expression for the total increment of energy in the 

 system due to any internal changes not involving any variation in 

 the total entropy or volume will consist of two parts, of which one 

 relates to the jn-operties of the masses of the system, and the other 

 may be expressed in the form 



62{0s), 

 the summation relating to all the surfaces of discontinuity. This 

 indicates the same tendency toward changes diminishing the value of 

 '2{ff s), which appears in other cases.* 



General Relations. — For any constant state of strain of the surface 

 of the solid, we may winte 



f/fs(i) — «^'/s(,)+/'2f^^^2(i)+/^3^^^'3(])+ etc., (674) 



since this relation is implied in the definition of the quantities 

 involved. From this and (659) we obtain 



da = - ?/s(i ) dt - /^2( I) ^A< 2 ~ ^ 3( 1) "^^/^s — ^t c, (675) 



which is subject, in sti'ictness, to the same limitation — that the state 

 of strain of the surface of the solid remains the same. But this 

 limitation may in most cases be neglected. (If the quantity a repre- 

 sented the true tension of the surface, as in the case of a surface 

 between fluids, the limitation would be wholly unnecessary.) 



Another method and notation. — We have so far supposed that we 

 have to do with a non-homogeneous film of matter between two 

 homogeneous (or very nearly homogeneous) masses, and that the 

 nature and state of this film is in all respects determined by the 



* The freezing together of wool and ice may be mentioned here. The fact that 

 a fiber of wool which remains in contact with a block of ice under water will become 

 attached to it seems to be strictly analogous to the fact that if a solid body be brought 

 into such a position that it just touches the free surface of water, the water will 

 generally rise up about the point of contact so as to touch the solid over a surface of 

 some extent. The condition of the latter phenomenon is 



fSA +o'wA>- (J^sw, 

 where the suffixes s, a, and w refer to the solid, to air, and to water, respectively. In 

 like manner, the condition for the freezing of the ice to the wool, if we neglect 

 the seolotropic properties of the ice, is 



<^sw + ^\\\ >■ f^si , 



whe'aa the suffixes s, W; and i relate to wool, to water, and to ice, respectively. See 

 Proc. Roy. Soc, vol. x, p. 447 ; or Phil. Mag., 4th ser., vol. xxi, p. 151. 



