Hence 



we 



have 



P = 

 V = 



dF , 



dF ( 

 ~~dJ ) 





where 





F = 



=iKU 2 



Ko-p 

 (3 



UJ + lKJ 2 ; . . . 



End osmose and other Allied Phenomena. 67 



(42) 



(43) 



that is, F possesses the characteristic property of a dissipation- 

 function*. If we had been entitled, a priori, to assert the 

 existence of such a function, the laws of the phenomena con- 

 sidered in § 4 could have been deduced from those of § 1. 



If the suffixes x and 2 refer to the circumstances of two 

 different experiments, we have 



Pi^+^j^p^+y^ (44) 



In particular if Pi=0, J 3 = 0, 



£*-£, • •• (45) 



as is otherwise evident from (41) and the preceding equations. 

 I do not know whether experiments on the electric trans- 

 fusion of liquids through a porous diaphragm, and on the 

 electromotive forces developed by difference of pressure between 

 the two sides, have ever been made with the same apparatus. 

 In any future experiments on these subjects, the testing of 

 the reciprocal relation (45) would be of interest, and would 

 apparently not present any great difficulty. 



Appendix. 



I give here the proofs of certain relations which hold be- 

 tween the fluid velocities u, v, w, and their space-derivatives 

 at any point of a rigid boundary. Some of these have been 

 employed in § § 1 and 4. 



Taking the origin on the boundary, and the axis of z along 

 the normal, let the equation to the boundary be 



s= i(A« 2 + 2Ba?y + Cy 2 ) + £(IV + 3G^ + SRccy* + % s ) + . (46) 



Let us first express the kinematical condition that the velocity 

 in the direction of the normal is zero at all points of the wall. 

 The direction-cosines of the normal at any point (x, y) near 



* See Rayleigh's ' Sound/ i. § 81. 

 F2 



