EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 145 



By these equations the general condition of equilibrium may be 

 reduced to the form 



ft SDn-fp SDv+ffr SDm t ... +//* SDm n 



+fg ShDm +fgh 8Dm l . . . +fgh SDm n > 0. (224) 



Now it will be observed that the different equations of condition 



affect different parts of this condition, so that we must have, 

 separately, 



ftSDri^Q, if fSDri = 0; (225) 



-fp SDv +fg ShDm ^ 0, (226) 



if the bounding surface is unvaried ; 



^Q, if /d-Dm^O; 



(227) 

 n^Q, if fSDm n =0. , 



From (225) we may derive the condition of thermal equilibrium, 



t = const. (228) 



Condition (226) is evidently the ordinary mechanical condition of 

 equilibrium, and may be transformed by any of the usual methods. 

 We may, for example, apply the formula to such motions as might 

 take place longitudinally within an infinitely narrow tube, terminated 

 at both ends by the external surface of the mass, but otherwise 

 of indeterminate form. If we denote by m the mass, and by v the 

 volume, included in the part of the tube between one end and a 

 transverse section of variable position, the condition will take the form 



-fp Sdv+fg Sh dm ^ 0, (229) 



in which the integrations include the whole contents of the tube. 

 Since no motion is possible at the ends of the tube, 



fp Sdv +JSv dp =fd(p Sv) = 0. (230) 



Again, if we denote by y the density of the fluid, 



fg Sh dm =fg ^Svydv =fgy Sv dh. (231) 



By these equations condition (229) may be reduced to the form 



fSv (dp +gy dh) ^ 0. (232) 



Therefore, since Sv is arbitrary in value, 



dp=-g-ydh, (233) 



which will hold true at any point in the tube, the differentials being 

 taken with respect to the direction of the tube at that point. There- 

 fore, as the form of the tube is indeterminate, this equation must hold 



true, without restriction, throughout the whole mass. It evidently 

 G.I. K 



