EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 281 



Arranging and combining terms, we have 



f(gy to + Sp) Dv +f[(p"-p') SN+ <T( CI + c 2 ) SN+gT Sz - fo] Da 



+/2(<r<$T)DJ = 0. (611) 



To satisfy this condition, it is evidently necessary that the coefficients 

 of Dv, Ds, and Dl shall vanish throughout the system. 



In order that the coefficient of Dv shall vanish, it is necessary and 

 sufficient that in each of the masses into which the system is divided 

 by the surfaces of tension, p shall be a function of z alone, such that 



In order that the coefficient of Ds shall vanish in all cases, it is 

 necessary and sufficient that it shall vanish for normal and for 

 tangential movements of the surface. For normal movements we 

 may write 



&r = and Sz 



where denotes the angle which the normal makes with a vertical 

 line. The first condition therefore gives the equation 



(613) 



which must hold true at every point in every surface of discontinuity. 

 The condition with respect to tangential movements shows that in 

 each surface of tension a- is a function of z alone, such that 



In order that the coefficient of Dl in (611) shall vanish, we must 

 have, for every point in every line in which surfaces of discontinuity 

 meet, and for any infinitesimal displacement of the line, 



2(<r<JT) = 0. (615) 



This condition evidently expresses the same relations between the 

 tensions of the surfaces meeting in the line and the directions of 

 perpendiculars to the line drawn in the planes of the various surfaces, 

 which hold for the magnitudes and directions of forces in equilibrium 

 in a plane. 



In condition (603), the variations which relate to any component are 

 to be regarded as having the value zero in any part of the system in 

 which that substance is not an actual component.* The same is true 



*The term actual component has been defined for homogeneous masses on page 64, 

 and the definition may be extended to surfaces of discontinuity. It will be observed 

 that if a substance is an actual component of either of the masses separated by a surface 

 of discontinuity, it must be regarded as an actual component for that surface, as well as 

 when it occurs at the surface but not in either of the contiguous masses. 



