64 Prof. H. Lamb on the Theory of Electric 



and if I be small in comparison with the radii of curvature 

 of the walls &c., we may neglect the second terms in the 

 brackets*. Under the same circumstances we shall also 

 have, approximately, 



du _, d 2 u ^ 



dx dzdx I 



dv = j^ \ (31) 



dy dzdy J 



so that the expression for the strength of the " source " 

 becomes 



/ d 2 u d 2 v \ 



' p \dxdz + dydzr 



pl d^' 



We may further neglect d 2 wfdx 2 , d 2 w/dy 2 in comparison 

 with dhvjdz 2 , so that the last expression may be written 



pl\/ 2 w, 

 which equals 



pi dp 



ft dz 



by (12). Hence (29) makes 



1 dd> 



source = f, 



a dz 



which is the proper surface condition for <f>. 



5. A similar investigation applies to the electromotive 

 forces called into play by the motion of solid particles through 

 a liquid. This phenomenon, 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 descending 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 <jpjfi 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 buoyaney) of the particles 

 vertically over the unit area. In Dorn's experiments tlx> 

 observed excess of potential was in fact positive, in accordanee 

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

 the data are not sufficient for further comparison with theory. 



* The justification of these and the following approximations is givep 

 in the Appendix. 



