ON ELECTRIC ENDOSMOSE AND OTHER ALLIED PHENOMENA. 505 



SO that the expression for the strength of the ' source ' becomes 



7 ( ^~'^ J- ^^^ ^ 

 \dxdz clydzj ' 

 or 



We may further neglect d^wjdx^, dhvjdy^ in comparison with d'^wjdz'^, 

 so that the last expression may be written 



which equals 



pi dp 



fx dz 

 by (12). Hence (29) makes 



source = -^, 



a dz 



which is the proper surface condition for ^. 



5. A similar investigation applies to the electromotive forces called 

 into play by the motion of solid particles through a liquid. This pheno- 

 menon, which is in a sense the converse of that discussed in § 3, has been 

 observed by Dorn in the case of grains of sand, or glass beads, descend- 

 ing by gravity through a vertical column of water. For the case of 

 steady motion the formula (29) shows that the top of the column will 

 be at a higher potential than the base by an amount equal to Tpj ji times 

 the pressure per unit area of the base due to the solid particles. This 

 pressure is equal to the effective weight (i.e., the gravity minus the buoy- 

 ancy) of the particles vertically over the unit area. In Dorn's experi- 

 ments the observed excess of potential was in fact positive, in accordance 

 with the general rule that p (and therefore E) is positive, but the data 

 are not suEBcient for further comparison with theory. 



The details of the process may be illustrated by the case of a spherical 

 particle. If r denote the distance from the centre, 6 the angular distance 

 from the lowest radius, the stream- function for the relative motion is of 

 the form 



* = ^^ + Br-iVr2) sin2^ . . . (32) 



where Y is the velocity of the sphere. The relative velocity of the fluid 

 over the surface is therefore 



if R be the radius. In consequence of the slipping, the zone bounded by 

 6 and O+dO gains electricity at the rate 



d 

 — pj^(27rRsin0. @)de. 



Dividing by the area 27rR2 sin 6 . dO of the zone, we find that each point of 

 the spherical surface is, in regard to the surrounding conducting mass, a 

 source of electricity of strength 



2/A B \ 



-rU^-r+^Jp^"^^ 



