60 Prof. H. Lamb on the Theory of Electric 



These phenomena have been explained in a general manner 

 by Quincke and von Helmholtz. If E denote the contact 

 difference of potential between the solid particle and the fluids 

 we have electrifications + cE on the opposed surfaces, which 

 are therefore urged in opposite directions by the electric 

 forces whose components are — d<f>/dx, —dfy/dy, — d<f>/dz. 



The principles of this paper lead to a very simple expression 

 for the velocity of an isolated particle when the motion has 

 become steady, viz., the velocity relative to the fluid in this 

 neighbourhood is in the direction of the electric current, and 

 its amount is 



V=-C P /£, (20) 



where C denotes the gradient of electric potential and p, 13, 

 have the same meanings as before. To prove this, take the 

 axis of # parallel to the general direction of the electric cur- 

 rent in the neighbourhood of the particle. The problem is 

 virtually unaltered if we suppose the fluid to flow with the 

 general velocity — Y past the solid, which is at rest. The 

 electric potential at a distance from the solid will be of the 

 form 



0=-Ctf + S o + S_ 2 + S_ 3 + , . . . (21) 



where S , S_2, S_3 .... are solid harmonics of the degrees 

 indicated. These latter terms represent the disturbance of 

 the otherwise uniform flow of electricity by the presence of 

 the insulating solid particles. It will be found that all the 

 conditions of our problem are satisfied by supposing the fluid 

 motion to be irrotational. We therefore write for the velocity 

 potential at a distance 



X =_V.* + T + T_ 2 + T_3 + . . . . (22) 



where T , T_ 2 , T_ 3 . . . are solid harmonics. The surface 

 condition will be of the form (13), in which we may neglect 

 the first term if we suppose the quantity I defined by (8) to 

 be small in comparison with the dimensions of the particle*. 

 Hence the condition is satisfied provided 



X=-%4>1 (23) 



and therefore 



V=-Cp//3 (24) 



In order to satisfy ourselves that the assumption (23) makes 



* For the case of a sphere of a radius R, I find without making this 

 approximation that 



V~-^(H-2J/K). 



