198 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



reference and in its variable state. (This involves no loss of generality, 

 since we may make the unit of length as small as we choose.) Let 

 the fluid meet the solid on one or both of the surfaces for which Z' 

 is constant. We may suppose these surfaces to remain perpendicular 

 to the axis of Z in the variable state of the solid, and the edges in 

 which y' and z' are both constant to remain parallel to the axis of X. 

 It will be observed that these suppositions only fix the position of 

 the strained body relatively to the co-ordinate axes, and do not in 

 any way limit its state of strain. 



It follows from the suppositions which we have made that 



dz _ dz _ dy 



-T-, = const. = 0, -j, const. = 0, -^ = const. = ; (398) 



and Z F =0, F z . = 0, Z z ,= -p^jjt. -, (399) 



Hence, by (355), 



dx 7 



dff. (400) 



Again, by (388), 



de = tdr] + T]dtpdv vdp+mdjUL 1 . (401) 



Now the suppositions which have been made require that 



dx dy dz 



V= M$M> < 402 > 



, 7 dy dz -.dx , dz dx 7 dy , dx dy 7 dz < ,.. 



and dv = -f-, -, -, d j- t -f T - f -T-? d -, -f -T-, -^- f d-r-, . (403) 



dy dz dx dz dx dy dx dy dz 



Combining equations (400), (401), and (403), and observing that 

 v , and r) y , are equivalent to e and TJ, we obtain 



dy dz\ -.dx , ^ -.dx , / T , dz dx\ 7 dy 



The reader will observe that when the solid is subjected on all sides 

 to the uniform normal pressure p, the coefficients of the differentials 

 in the second member of this equation will vanish. For the expression 



-p> -7-7 represents the projection on the Y-Z plane of a side of the 



parallelepiped for which x r is constant, and multiplied by p it will 

 be equal to the component parallel to the axis of X of the total 

 pressure across this side, i.e., it will be equal to X x > taken negatively. 



The case is similar with respect to the coefficient of d-p,; and X?, 

 evidently denotes a force tangential to the surface on which it acts. 



