EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 189 



We may therefore use (361) to eliminate the fourth and fifth integrals 

 from (360). If we multiply (362) by p, and take the integrals for 

 the whole surface of the solid and for the fluid in contact with it, we 

 obtain the equation 



f*p 8Dv = -fp(a8x+/3Sy + ySz)D 8 -fpv v , WDa', (365) 



by means of which we may eliminate the sixth integral from (360). 

 If we add equations (363) multiplied respectively by yu 1? yu 2 , etc., 

 and take the integrals, we obtain the equation 



(366) 



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

 The condition of equilibrium is thus reduced to the form 



+f v ,8N'Ds'-ftn v ,SN'Ds'+fp(a8x+/3Sy+ 7 Sz)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 fa , 



/* 2 > etc - 



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



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

 of the solid in the state of reference, we have 



X* Sx Ds' -fff^-j. Sxdx'dy'dz', (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 



8'^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 



