348 J. W. (jrlbbs — J^qallibrlmit of Heterogeneous /Substa/ices. 



2,fy,dDm, = -/^,(A/.r/) 6]>^'Bs', (366) 



by means of whicli we may eliminate the last integral from (360). 

 The condition of eqnilil)rium is thus reduced to the form 



/ff2 2' ( JTx, S '^,) dx' dy' dz' +///</ F' Sz d^ dy' dz' 



+ J'£v, <S-^' Ds' -ft ;/v, dW I)s' -^fp {a6x + fi 6y + y 6z) Ds 



J^fpvy,6N' Bs' -f2^ i/tj\') dJ^'Ds'^0, (367) 

 in which the variations are independent of the equations of condition, 

 and in which the only quantities relating to the fluids are p and //j, 

 yWg, etc. 



Now by the ordinary method of the calculus of variations, if we 

 write a', fi', y' for the direction-cosines of the normal to the surface 

 of the solid in the state of reference, we have 



fffX^.d'-^^dx'dy'dz' 



=fa' X^, 6x I)s' - ff/^^^ ^x d-^' <^m ilz\ (368) 



with similar expressions for the other parts into which the first 

 integral in (367) may be divided. The condition of equilibrium is 

 thus reduced to the form 



-fff^ 2' (^^ 6x^ dx' dy' dz' +fff!/ r Sz dx! dy' dz' 



-\-f:^ ^' («' JQ, Sx) Ds -i-fp 2 {a dx) Bs 



-i-f[£y, - t //v, +i> Vy, - ^\ (/I, / V)] S^' I^s'^ 0. (369) 

 It must be observed that if the solid mass is not continuous 

 throughout in nature and state, the surface-integral in (368), and 

 therefore the first surface-integral in (369), must be taken to apply 

 not only to the external surface of the solid, but also to every surface 

 of discontinuity within it, and that with reference to each of the 

 two masses separated by the surface. To satisfy the condition of 

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

 throughout the solid mass 



2 2' (^^^-^V^^) - ff r Sz =0- (370) 



that throughout the surfaces where the solid meets the fluid 



Bs' 2 2' {a' X^, Sx) -f Bsp 2 {a Sx) = 0, (37 1) 



and 



[ey,^t7fy,+pvy,^2,{/j,l\')]SX'^0; (372) 



and that througliout the internal surfaces of discontinuity 



