162 



Mr. W. M. Hicks on 



[June 19, 



V. " The Motion of Two Spheres in a Fluid." By^W. M. Hicks, 

 M.A., St. John's College, Cambridge. Communicated by 

 Professor J. Clerk Maxwell, F.R.S., Professor of Ex- 

 perimental Physics in the University of Cambridge. 

 Received May 16, 1878. 



(Abstract.) 



In the investigation the method of images is followed, and is based 

 on the following lemma. The " image " of a source in an infinite fluid 

 in presence of a sphere consists of a source at the inverse point of the 

 former, and a line sink thence to the centre of the sphere. If the 



original source be jjl at P, the inside sources ^p-/"? and the line density 



of the sink= — --, a being the radius of the sphere. For the analogous 



a 



case of fluid within a sphere we, of course, require to have an equal 

 source and sink somewhere within, else there must ensue motion 

 across the boundary. In this case the analysis would give an infinite 

 term for a single source, which, when the quantity of source and sink 

 is zero, disappears. The image for a source would be as before a source 



^pf 1 a ^ the inverse point of P, and a line sink — — , thence to infinity. 



It is this last which produces the infinite term. From these lemmas 

 are deduced the images of " doublets," by which are meant the singular 

 points resulting from the coalescence of an equal source and sink whose 

 magnitude varies inversely as the distance between them. It is easily 

 seen that when the axis of a doublet passes through the centre of the 

 sphere, its image is a single doublet within it, at the inverse point of 



the former of magnitude — (^jj^ /*» p being the strength of the original 



doublet. A similar result holds for the image of a doublet within a 

 sphere, whose axis passes through the centre of the sphere. 



The image of a doublet whose axis is perpendicular to the line to the 



centre is not so simple. It consists as before of a doublet (jy^ ** a ^ the 



inverse point, bnt besides this there stretches from the inverse point to 

 the centre a negative line doublet of variable density, the line density 



at a distance, r, being — -gp-- The only difference between this and the 



case for an internal doublet is that the corresponding line doublet 

 stretches to infinity. 



No attempt is made to find the velocity potentials, but supposing 

 them known, the kinetic energy of the motion for two spheres is 



