352 J. W. Gibbs — Equilibrmm of Heterogeneous Substances. 



versa). Now i' is a right-angled triangle of which the perpendicular 

 sides raay be called di/' and dz'. The projection of L on the V-Z 

 plane will be a triangle, the angular points of which are determined 

 by the co-ordinates 



y, z; y-rf^-,dy\ ^ + 7^-'^^; y+^<^^' ^ + "7/^^^' 



the area of such a triangle is 



'\dy''dz' dy' dz'J ■' ' 

 or, since h dy' dz' represents the area of //', 



\dy' dz' dy' dz'J 

 (That this expression has the proper sign will appear if we suppose 

 for the moment that the strain vanishes.) The areas of the pro- 

 jections of TJf and iVupon the same plane will be obtained by chang- 

 ing y', z' and a' in this expression into z', cc', and /^', and into x', y', 

 and y'. The sum of the three expressions may be substituted for 

 aDs'm (381). 



We shall hereafter use '2' to denote the sum of the three terms 

 obtained by rotary substitutions of quantities relating to the axes 

 X', Y\ Z', (i. e., by changing x\ y', z' into y', z, x', and into z', x', y', 

 with similar changes in regard to a', p', y', and other quantities 

 relating to these axes,) and 2 to denote the sum of the three terms 

 obtained by similar rotary changes of quantities relating to the axes 

 X, Y, Z. This is only an extension of our previous use of these 

 symbols. 



With this understanding, equations (381) may be reduced to the 

 form 



, ,- s ^., ( , / dy dz dz dy\) ^ ^ 



2' (a' Ax,) +?) 2' \ a'{^,-yr ^, -/,- U = 0, 



V \o-i-J I \dy' dz' dy' dz' / \ ). (382) 



etc. J 



The formula (372) expresses the additional condition of equilibrium 

 which relates to the dissolving of the solid, or its growth without 

 discontinuity. If the solid consists entirely of substances which are 

 actual components of the fluid, and there are no passive resistances 

 which impede the formation or dissolving of the solid, dN' may have 

 either positive or negative values, and we must have 



Sy, - tfjy,^pv,„=2^{ixj\'). (383) 



But if some of the components of the solid are only ])ossible com- 



