502 REPORT— 1887. 



^ = _Ca- + So + S_2+S_3+ . . . (2i) 



where So, S_2, S_3 . . . are solid harmonics of the degrees indicated. 

 These latter terms i-epresent 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=_\ra; + To + T_2 + T_3+ . . . (22) 



where To, T.,, 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 dimen- 

 sions 01 the particle.^ Hence the condition is satisfied provided 



X = -f.^ .... (23) 



and therefore 



Y=-Cpli5 .... (24) 



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

 ant force and couple on the sphere equal to zero, it will be suSicient to 

 show that the force and couple-resultants of the stress across a closed 

 surface 2i drawn in the fluid and just enclosing the solid are zero. Using 

 a common notation for the components of stress at any point of the fluid 

 we have 



P.r.,= -P + 2/7-§, &C., &C.| 



JO '-' [ . . . (25) 



P,~. = 2/' j^_^, &c., &c. ) 



where p is constant, by (12). The resultant stress parallel to x across 

 the complete boundary 2 of any space occupied by fluid is 



J J (h'-r.:: + '"P-rj, + «Px.) d^, 



where Z, «i, n are the direction-cosines of the normal to any element c?2 of 

 the boundary. This surface-integral is equal to the volume-integral 



lfl(%-%+%')"»^^ 



taken throughout the interior of 2, which vanishes, by (25), since ^^^=0. 

 In a similar manner it may be shown that the couple-resultant of the 

 stress across 2 is zero. Now let 2 be made up of the surface 2i above 

 defined, and of a sphere 2^ of infinite radius having its centre at the 

 origin. It follows that the stresses across 2i are statically equivalent to 

 those across 22- And it easily follows from (22) that the latter stresses 

 are in equilibrium. 



It is remarkable that the velocity (24) is independent of the size or 

 shape of the particle, so long as its dimensions are large in comparison 

 with I. This velocity is, of course, to be superposed on that of the fluid 



' For the case of a sphere of radius R, I find without making this approximation that 



