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XIII. On the Motion of Two Spheres in a Fluid. 
By W. M. Hicks, M.A., Fellow of St. Johns College, Cambridge. 
Communicated by Professor J. Clerk Maxwell, F.R.S. 
Received May 10—Read June 19, 1879. 
The general theory of the motion of a single rigid body through an infinite incom¬ 
pressible fluid is well known, chiefly through the work of Thomson and Tait* and 
KiRCHHOFF,t and we are able to calculate numerically the results in the case of the 
sphere, the ellipsoid, and a large number of cylindrical surfaces. The theory of the 
motion of two or more bodies in a fluid has naturally not made the same progress, 
and we are unable to determine the form of the expressions involved for the general 
motion of any particular solids. So far as I am aware, the first attempt was made by 
Stokes, in a paper read before the Cambridge Philosophical Society in 1843, entitled 
“On some cases of Fluid Motion.In this paper, amongst other problems, he con¬ 
siders the case of two spheres. He determines the instantaneous velocity potential 
for two concentric spheres and for two concentric cylinders with fluid between them, 
and finds that the effect of the fluid is to increase the inertia of the inner sphere by 
, a 3 -f25 3 
a mass =h.~n -r 
* b 6 — n- 
of the mass of the fluid displaced, and that of the inner cylinder 
1 ^ 2 _|_ ^2 
by a mass ——— of the mass displaced, a, b, being the radii of the spheres or cylinders. 
He also approximates to the cases where one sphere is moving in the presence of 
another in an infinite fluid; and also in the presence of a plane, the method used 
being first to calculate the velocity potential for any motion of the points of the 
plane, and then suppose them actually animated with velocities equal and opposite 
to the normal velocities of the fluid motion at those points if the plane had been 
removed. He applies the same method also to the consideration of the motion of 
two spheres. In a note in the Report of the British Association at Oxford, 1847, 
he states the theorem given by me in § 4 relating to the image of a doublet whose 
axis passes through the centre, and mentions that this will easily serve to determine 
the motion. In 1863 Herr Bjerknes communicated a paper to the Society of 
Sciences at Christiana, on the motion of a sphere which changes its volume, and in 
* Nat. Phil., p. 264, new edition, p. 330. 
t Bokchardt, Bd. 71. 
t Carnb. Phil. Trans., vol. viii. 
3 N 2 
