of Mixed Solutions. 279 



and by means of (4) the equilibrium condition can be expressed 



8ff[(ir' + pU+P)dxdydz=0, 



or introducing the thermodynamic potential for constant pressure co' 



s(l{ (co' + pU+P-p) dxdydz = 0. 



The variation must be subject to the condition that no mass 

 leaves the boundary. Putting 



(11) c'^-pU+P-p=W, 



we get the equilibrium conditions 



(12a) 8 \\\w dxdydz = 0, 



( 1 26) S Ului dxdydz = 0, i = 0, 1, 2 . . . r. 



(12c) 8 pdxdydz =0. 

 (12c) is of course a consequence of (126). 



§ 4. Solution of the problem of variation. 



We shall now specify our bounding surface; and, as we have 

 seen, this can be done without restricting the generality of the 

 solution. We shall consider the fluid inside a parallelopipedic 

 element {/^x. Ay, Az), and, for the sake of simplicity, one of the 

 corners may be at the origin of the coordinates. Further, we shall 

 let the element have a needle form, or we put 



Ay = aaAx'>^\ Az = a^Ax""^, 



where a^ and a.^ are finite quantities and Wj and n^ are positive 

 numbers, which can be given any values we like, and thus we can 

 always be able to put 



%i 5 2 and no ^ 2. 



The two ends of the element we shall denote by (0) and (1). The 

 element is supposed to have rigid boundaries except at the end, 

 which consists of the piston exerting a pressure P. 



We shall specify the variation, which consists in a relative 

 displacement of masses inside the element, by supposing that 

 the masses are suffer a translation parallel to the principal axis of 

 the element. Under these conditions the equilibrium conditions 

 take the form 



VOL, XV. PT. III. 19 



