ON ELECTBIC EXDOSMOSE AND OTHER ALLIED PHENOMENA. 507 



Again, in the case of § 4 we liave J = 0, and therefore 



V= AU = ^- . P. 

 K 



The results we have obtained show that 



K = A.= -Ko-p//3 .... (41) 

 Hence we have 



p=^ 1 



'i \ (*2) 



dJ J 

 where 



F = iKU2-^^UJ + |RP . . . (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 tlie phenomena considered in § 4 could have been deduced 

 from those of § 1. 



If the suffixes j and 2 refer to the circumstances of two different 

 experiments we have 



PiU2+V,J,=P2U,+V,Ji . . . (44) 



In particular if Pi" 0, J..=0, 



^•--^^ (45) 



P., J, ^ ^ 



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



I do not know whether experiments on the electric transfusion 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 proof of certain relations which held between the fluid 

 velocities u, v, iv, 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 



z=i,{kx^ + 2Bx>j + Cy^)+i(Fx^ + mx'-y + 3Bxi/- + K7/)+ . . (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 («, 2/) near the origin are 



-(kx + By)-^(¥x'' + 2Gxy + m/) ) 

 -(Bx + Cy)-l.{Gx' + 2my + Ky'-) , . (47) 



l-^(Ax + Byy-^(Bx + Cyy ) 



' See Eayleigh's Sound, i. § 81. 



