456 
MR. W. M. HICKS OK THE MOTION OF TWO SPHERES IN A FLUID. 
which he approximates for the motion of two spheres. I have not been able to see this 
paper, nor some others which he presented to the same Society at some later periods ; 
but he has given an account of his researches in the ‘ Comptes Rendus/'* together 
with some historical notices on the development of the theory. He does not seem, 
however, to have been acquainted with the important paper of Stokes above referred 
to.t In 1867 Thomson and Tait’s ‘ Natural Philosophy ’ appeared, containing general 
theorems on the motion of a sphere in a fluid bounded by an infinite plane, viz.: that 
a sphere moving perpendicularly to the plane moves as if repelled by it, whilst if it 
moves parallel to it it is attracted. In a paper on vortex motion in the same year 
(Edin. Trans., vol. xxv.), Thomson proved that a body or system of bodies passing on 
one side of a fixed obstacle move as if attracted or repelled by it, according as the 
translation is in the direction of the resultant impulse or opposite to it. In the 
‘Philosophical Magazine’ for June, 1871, Professor Guthrie publishes some letters 
from Sir W. Thomson on the apparent attraction or repulsion between two spheres, 
one of which is vibrating in the line of centres. Results only are given, and he states 
that if the density of the free globe is less than that of the fluid, there is a “ critical ” 
distance beyond which it is attracted, and within which it is repelled. The problem 
of two small sjiheres is also considered by Kirchhoef in his ‘ Vorles. ti. Math. Phys.,’ 
pp. 229, 248. In his later papers Bjerknes takes up the question of “ pulsations ” as 
well as vibrations. Of solutions for other cases than spheres, Ktrchhoff has con¬ 
sidered!. the case of two thin rigid rings, the axes of the rings being any closed 
* ! Comptes Rendus,’ tom. Ixxxiv., p. 1222, &c. 
t Not only Herr Bjerknes, but several writers on the Continent seem to be unacquainted with this 
paper of Stokes, and also with Green’s papers. Kirchhoef, in his ‘Vorlesungen uber Mathematische 
Physik ’ (second edition, p. 227), says that Dirichlet first treated the motion of a sphere in a fluid in the 
Monatsberichte der Berk Akad.’ in 1852, and Clebsch that of the ellipsoid in 1856, in ‘ Crelle,’ Bd. 52. 
Bjerknes also repeats the same statement, and Clebsch in his paper regards Dirichlet as the first to 
solve for the sphere. In his paper Dirichlet says: “ Wie es scheint, ist bis jetzt fiir keinen noch so 
einfacken Fall der Widerstand, den ein in einer ruhenden Fliissigkeit fortbewegter fester Korper von 
dieser erleidet, aus den seit Euler bekannten allgemeinen gleichungen der Hydrodynamik abgeleitet 
worden.” The fact is that Green in a paper read before the Royal Society of Edinburgh in 1833, 
entitled “ Researches on the Vibrations of Pendulums in Fluid Media ” (Trans. Roy. Soc. Edin. ; also 
published in the Reprint of his papers, p. 313), and written without the knowledge of Poisson’s paper of 
1831, “ Sur les mouvements simultanes d’un pendule et de l’air environnant,” treated of the motions of 
an ellipsoid moving parallel to one of its axes. He obtains the velocity potential as an elliptic integral 
for a motion parallel to an axis, which also of course contains implicitly that for the sphere. He shows 
that it is necessary to suppose the density of the body augmented by a quantity proportional to the 
density of the fluid. For the case of the spheroids moving in their equatorial planes or parallel to their 
axes he completely determines this quantity, whilst for the sphere he finds that it is one-half the mass of 
the fluid displaced. The first place in which I have been able to find the well known form of the velocity 
potential for a sphere is in Stokes’ paper of 1843 before mentioned. He obtains it as a particular case of 
a more general problem, and refers to it as the “known” value for the sphere. The equations of the 
lines of flow were, I believe, first given by Dirichlet. 
t Borchardt, Bd. 71, 
