16 REPORT—1846. 
Magazine (New Series). In the paper just referred to, Prof. Challis finds that 
when the fluid is incompressible there is no decrement in the arc of oscilla- 
tion, except what arises from friction and capillary attraction. In the case 
of air there is a slight theoretical decrement; but it is so small that Prof. 
Challis considers the observed decrement to be mainly owing to friction. 
This result follows also from Poisson’s solution. Prof. Challis also finds that 
a small sphere moving with a uniform velocity experiences no resistance, and 
that when the velocity is partly uniform and partly variable, the resistance 
depends on the variable part only. The problem, however, referred to in 
the title of this paper, is that of calculating the motion of a small sphere 
situated in an elastic fluid, and acted on by no forces except the pressure of 
the fluid, in which an indefinite series of plane condensing and rarefying 
waves is supposed to be propagated. This problem is solved by the author on 
principles similar to those which he has adopted in the problem of an oscil- 
lating sphere. The views of Prof. Challis with respect to this problem, which 
he considers a very important one, are briefly stated at the end of a paper 
published in the Philosophical Magazine*. 
In a paper ‘On some Cases of Fluid Motion,’ published in the Trans- 
actions of the Cambridge Philosophical Society+, I have considered some 
modifications of the problem of the ball pendulum, adopting in the main the 
principles of Poisson, of the correctness of which I feel fully satisfied, but 
supposing the fluid incompressible from the first. In this paper the effect of 
a distant rigid plane interrupting the fluid in which the sphere is oscillating is 
given to the lowest order of approximation with which the effect is sensible. 
It is shown also that when the ball oscillates in a concentric spherical enve- 
lope, the effect of the resistance of the fluid is to add to the mass of the 
= 5° where a is the radius of the ball, 6 that 
of the envelope, and m the mass of the fluid displaced. Poisson, having 
reasoned on the very complicated case of an elastic fluid, had come to the 
conclusion that the envelope would have no effect. 
One other instance of fluid motion contained in this paper will here be 
mentioned, because it seems to afford an accurate means of comparing theory 
and experiment in a class of motions in which they have not hitherto been 
compared, so far as I am aware. When a box of the form of a rectangular 
parallelepiped, filled with fluid and closed on all sides, is made to perform small 
oscillations, it appears that the motion of the box will be the same as if the 
fluid were replaced by a solid having the same mass, centre of gravity, and 
principal axes as the solidified fluid, but different principal moments of in- 
ertia. These moments are given by infinite series, which converge with 
extreme rapidity, so that the numerical calculation is very easy. The oscil- 
lations most convenient to employ would probably be either oscillations by 
torsion, or bifilar oscillations. 
VI. M. Navier was, I believe, the first to give equations for the motion of 
fluids without supposing the pressure equal in all directions. His theory is 
contained in a memoir read before the French Academy in 1822{. He con- 
siders the case of a homogeneous incompressible fluid. He supposes such a 
fluid to be made up of ultimate molecules, acting on each other by forces 
which, when the molecules are at rest, are functions simply of the distance, 
but which, when the molecules recede from, or approach to each other, are 
modified by this cireumstance, so that two molecules repel each other less 
sphere a mass equal to 
strongly when they are receding, and more strongly when they are approaching, - 
* Vol. xviii. New Series, p. 481. t Vol. viii. p. 105. 
t Mémoires de l’Académie des Sciences, tom. vi. p. 389. 
