190 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



equilibrium, as thus understood, it is necessary and sufficient that 

 throughout the solid mass 



ISf(2jjZto)-grto-0; (370) 



that throughout the surfaces where the solid meets the fluid 



JV2ZV^x'to)+#*l>2(a&0 = 0, (371) 



and [v'-fyv'+l>iV-2 1 i r i ')] SN'^0 ; (372) 



and that throughout the internal surfaces of discontinuity 



where the suffixed numerals distinguish the expressions relating to 

 the masses on opposite sides of a surface of discontinuity. 



Equation (370) expresses the mechanical conditions of internal 

 equilibrium for a continuous solid under the influence of gravity. If 

 we expand the first term, and set the coefficients of Sx, Sy, and Sz 

 separately equal to zero, we obtain 



(374) 



dX z >_ 



' ~ ' ~ ' 



dx' ~ dy' ~ dz 



x , dY T dY z ,_ 



~ ' 



dx dy dz 

 dZ, 



dx' dy' dz' 



The first member of any one of these equations multiplied by dw'dy'dz' 

 evidently represents the sum of the components parallel to one of the 

 axes X, F, Z of the forces exerted on the six faces of the element 

 dx'dy'dz' by the neighboring elements. 



As the state which we have called the state of reference is arbitrary, 

 it may be convenient for some purposes to make it coincide with the 

 state to which x, y, z relate, and the axes X', F, Z with the axes 

 X, F, Z. The values of X %>,... Z z > on this particular supposition 

 may be represented by the symbols X x , ... Z z . Since 



j 



dx' 



and since, when the states, x, y, z and x' y' z coincide, and the axes 



dx d\i 



X, F, Z, and X', F", Z', d-^, and d-^-, represent displacements which 



differ only by a rotation, we must have 



* r =F X) (375) 



and for similar reasons, 



Yz = Z Y , Z X = X 2 . (376) 



