OF FLUIDS ON THE MOTION OF PENDULUMS. [11] 



the principal axes of the ellipsoid become equal, the integral admits of expression in finite 

 terms, by means of circular or logarithmic functions. When the ellipsoid becomes a sphere, 

 Green's result reduces itself to Poisson's. 



In a memoir read before the Royal Academy of Turin on the 18th of January 1835, 

 and printed in the 37th Volume of the memoirs of the Academy, M. Plana has entered at 

 great length into the theory of the resistance of fluids to pendulums. This memoir contains, 

 however, rather a detailed examination of various points connected with the theory, than the 

 determination of the resistance for any new form of pendulum. The author first treats the 

 case of an incompressible fluid, and then shews that the result would be sensibly the same in 

 the case of an elastic fluid. In the case of a ball pendulum, the only one in which a com- 

 plete solution of the problem is effected, M. Plana's result agrees with Poisson's. 



In a paper read before the Cambridge Philosophical Society on the 29th of May 1843, 

 and printed in the 8th Volume of the Transactions, p. 105, I have determined the resistance 

 to a ball pendulum oscillating within a concentric spherical envelope, and have pointed out 

 the source of an error into which Poisson had fallen, in concluding that such an envelope 

 would have no effect. When the radius of the envelope becomes infinite, the solution agrees 

 with that which Poisson had obtained for the case of an unlimited mass of fluid. I have 

 also investigated the increase of resistance due to the confinement of the fluid by a distant 

 rigid plane. The same paper contains likewise the calculation of the resistance to a long 

 cylinder oscillating in a mass of fluid either unlimited, or confined by a cylindrical envelope, 

 having the same axis as the cylinder in its position of equilibrium. In the case of an un- 

 confined mass of fluid, it appeared that the effect of inertia was the same as if a mass equal 

 to that of the fluid displaced were distributed along the axis of the cylinder, so that n = 2 

 in the case of a pendulum consisting of a long cylindrical rod. This nearly agrees with 

 Baily's result for the long 1^ inch tube; but, on comparing it with the results obtained with 

 the cylindrical rods, we observe the same sort of discrepancy between theory and observation 

 as was noticed in the case of spheres. The discrepancy is, however, far more striking in the 

 present case, as might naturally have been expected, after what had been observed with 

 spheres, on account of the far smaller diameter of the solids employed. 



A few years ago Professor Thomson communicated to me a very beautiful and powerful 

 method which he had applied to the theory of electricity, which depended on the consideration 

 of what he called electrical images. The same method, I found, applied, with a certain modi- 

 fication, to some interesting problems relating to ball pendulums. It enabled me to calculate 

 the resistance to a sphere oscillating in presence of a fixed sphere, or within a spherical enve- 

 lope, or the resistance to a pair of spheres either in contact, or connected by a narrow rod, 

 the direction of oscillation being, in all these cases, that of the line joining the centres of the 

 spheres. The effect of a rigid plane perpendicular to the direction of motion is of course 

 included as a particular case. The method even applies, as Professor Thomson pointed out 

 to me, to the uncouth solid bounded by the exterior segments of two intersecting spheres, 

 provided the exterior angle of intersection be a submultiple of two right angles. A set of 

 corresponding problems, in which the spheres are replaced by long cylinders, may be solved 

 in a similar manner. These results were mentioned at the meeting of the British Association 



26—2 



