ON ELECTEIC ENDOSMOSE AMD OTHER ALLIED PHENOMENA. 499 



e.g., the experiments of Poiseuille. The slipping leads to appreciable 

 results in the cases at present in view, only in consequence of the relatively 

 enormous electrical forces acting on the superficial film, and dragging the 

 fluid (as it were) by the skin, through the tube. 

 The formula (6) may be written— 



Flux of liquid crcE 



Flux of electricity /3 ' ' * ' ^ ' 



In this form it can be shown to be true, under a certain restriction, for a 

 tube of varying section, for a network of tubes, and even for the labyrinth 

 of channels contained in the walls of a porous vessel, provided no 

 difference of pressure be allowed to establish itself on the two sides. 



Let (f) denote as before the electric potential at any point of the fluid 

 It will appear that all the conditions of our problem will be satisfied if 

 ■we suppose the motion of the fluid to be irrotational, the velocity-potential 

 X being everywhere proportional to </>. 



Since v'-^x = ^' *'^® equations of steady small motion of a viscous 

 liquid, viz. — 



_J+,V^. = 



dp 

 'dy 



dp 

 ' dz 



+ juv'^w = 



(12) 



are satisfied by |? = const. To form the boundary condition correspond- 

 ing to (1), let ds be a linear element drawn on the surface in the direction 

 of the flow of liquid, and therefore also of electricity. We obtain — 



''/-/'|-4t=0 (13) 



where / is the rate of shear in a plane through ds normal to the surface. 

 If Z be small in comparison with the linear dimensions of the channels 

 the first term of this equation may, in the cases at present under con- 

 sideration, be neglected in comparison with the rest,' so that (13) is 

 satisfied provided — 



X=-|«^ . . . . . (14) 



everywhere. Hence the flow of liquid is everywhere in the same 



' To see this, take the orig-in at any point of the boundary, and the axis of z along 

 the normal, and let the equation to the boundary then be 



2 = i (Aa!= + 2Bxy + Cy-) + &c. 



If the axis of x be in the direction of the flow at 0, we have to prove that M^xM'^<i^ 

 may be neglected in comparison with dxjdx. It is proved in the appendix to this 

 paper that at we must have 



dw 



and therefore 



dxdz ' dx' 



which proves the statement made above, when I is small in com^ arisen with the 

 radii of curvature of the walL 



K E 2 



