OF FLUIDS ON THE MOTION OF PENDULUMS. [23] 



Section II. 



Solution of the equations in the case of a sphere oscillating in a mass ofjluid either 

 unlimited, or confined by a spherical envelope concentric with the sphere in its position of 

 equilibrium. 



9. Suppose the sphere suspended by a fine wire, the length of which is much greater 

 than the radius of the sphere. Neglect for the present the action of the wire on the fluid, and 

 consider only that of the sphere. The motion of the sphere and wire being supposed to take 

 place parallel to a fixed vertical plane, there are two different modes of oscillation possible. 

 We have here nothing to do with the rapid oscillations which depend mainly on the rotatory 

 inertia of the sphere, but only with the principal oscillations, which are those which are 

 observed in pendulum experiments. In these principal oscillations the centre of the sphere 

 describes a small arc of a curve which is very nearly a circle, and which would be rigorously 

 such, if the line joining the centre of gravity of the sphere and the point of attachment of the 

 wire were rigorously in the direction of the wire. In calculating the motion of the fluid, we 

 may regard this arc as a right line. In fact, the error thus introduced would only be a small 

 quantity of the second order, and such quantities are supposed to be neglected in the investi- 

 gation. Besides its motion of translation, the sphere will have a motion of rotation about a 

 horizontal axis, the angular motion of the sphere being very nearly the same as that of the 

 suspending wire. This motion, which would produce absolutely no effect on the fluid according 

 to the common theory of hydrodynamics, will not be without its influence when friction is taken 

 into account ; but the effect is so very small in practical cases that it is not worth while to take 

 it into account. For if a be the radius of the sphere, and I the length of the suspending wire, 

 the velocity of a point in the surface of the sphere due to the motion of rotation will be a 

 small quantity of the order al~ l compared with the velocity due to the motion of translation. 

 In finding the moment of the pressures of the fluid on the pendulum, forces arising from these 

 velocities, and comparable with them, have to be multiplied by lines which are comparable 

 with a, I, respectively. Hence the moment of the pressures due to the motion of rotation of 

 the sphere will be a small quantity of the order a 2 /" 8 , compared with the moment due to the 

 motion of translation. Now in practice I is usually at least 20 or 30 times greater than a, and 

 the whole effect to be investigated is very small, so that it would be quite useless to take 

 account of the motion of rotation of the sphere. 



The problem, then, reduces itself to this. The centre of a sphere performs small periodic 

 oscillations along a right line, the sphere itself having a motion of translation simply : it is 

 required to determine the motion of the surrounding fluid. 



10. Let the mean position of the centre of the sphere be taken for origin, and the 

 direction of its motion for the axis of w, so that the motion of the fluid is symmetrical with 

 respect to this axis. Let •sr be the perpendicular let fall from any point on the axis of x, q 



