J. TF. Gibhs — EquiUbriuin of Heterogeneous ^ubstancea. 447 



-J\P' -P") ^^'"^s -j-fSpDv +/0- (c, +C2) dJSTBs 

 +/2{(rST)Dl —/dffI>s+fgydzI)v+fc/r6zJJsz= 0. 



Arranging and combining terms, we have 



fig y6z-\- dp) Dv 



-\-f[{p"-p')S]^+ff{c,-{-c^)dJV+gr6z-S<r]IJs 



+ f^{adT)Dl=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, 2^ shall be a function of z alone, such that 



^=-gr- (612) 



az 



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 tan- 

 gential movements of the surface. For normal movements we may 

 write 



da =z 0, and 6z = cos B 6JV, 



where 5 denotes the angle which the normal makes with a vertical 

 line. The first condition therefore gives the equation 



p'—j)" = (T{e,-{-e,)-\-grcos^, (613) 



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

 The condition with respect to tangentiakmovements shows that in 

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



^=ffr. (6,4) 



dz 



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

 must have, for eveiy point in every line in which surfaces of discon- 

 tinuity meet, and for any infinitesimal displacement of the line, 



2(ff6T)=:0. (615) 



This condition evidently expresses the same relations between the ten- 

 sions of the surfaces meeting in the line and the directions of per- 

 pendiculars 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 



Trans. Conn. Acad., Vol. III. 57 Jan., 1878. 



