EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 193 



The projection of L on the Y-Z plane will be a triangle, the angular 

 points of which are determined by the co-ordinates 



dy j, . dz , , 



y, z; y, 

 the area of such a triangle is 



_ __ , , , , 

 dy'dz f dy'dz~'r y( 



or, since J dy r dz represents the area of L', 



(dy dz__dz dy\ , n , 

 \dy' dz' dy' dz') a 



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

 for the moment that the strain vanishes.) The areas of the projections 

 of M and N upon the same plane will be obtained by changing y f , z' 

 and a' in this^expression into 2', x', and /3', and into x', y', and y. The 

 sum of the three) expressions may be substituted for a Ds in (381). 



We shall hereafter use S' 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 y', z' into y', z', x', and into /, x', y r , 

 with similar changes in regard to a', fl', 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 



Y c^ 2/ dz dz dy\\_ 



a --- 



(382) 

 etc. 



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, SN' may have 

 either positive or negative values, and we must have 



v tffr, pv v , = Sj ( ywJV). (383) 



But if some of the components of the solid are only possible com- 

 ponents (see page 64) of the fluid, SN' is incapable of positive values, 

 as the quantity of the solid cannot be increased, and it is sufficient 



for equilibrium that 



e v , _ tl fr + pv , ^ 2/^iy). (384) 



To express- condition (383) in a form independent of the state of 



reference, we may use e v > ^v* I\, etc., to denote the densities of 

 G.I. N 



