X. On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. 

 By G. G. Stokes, M.A., Fellow of Pembroke College, and Lucasian Pro- 

 fessor of Mathematics in the University of Cambridge. 



[Read December 9, 1850.] 



The great importance of the results obtained by means of the pendulum has induced 

 philosophers to devote so much attention to the subject, and to perform the experiments with 

 such a scrupulous regard to accuracy in every particular, that pendulum observations may 

 justly be ranked among those most distinguished by modern exactness. It is unnecessary here 

 to enumerate the different methods which have been employed, and the several corrections 

 which must be made, in order to deduce from the actual observations the result which would 

 correspond to the ideal case of a simple pendulum performing indefinitely small oscillations 

 in vacuum. There is only one of these corrections which bears on the subject of the present 

 paper, namely, the correction usually termed the reduction to a vacuum. On account of 

 the inconvenience and expense attending experiments in a vacuum apparatus, the observations 

 are usually made in air, and it then becomes necessary to apply a small correction, in order 

 to reduce the observed result to what would have been observed had the pendulum been 

 swung in a vacuum. The most obvious effect of the air consists in a diminution of the moving 

 force, and consequent increase in the time of vibration, arising from the buoyancy of the 

 fluid. The correction for buoyancy is easily calculated from the first principles of hydro- 

 statics, and formed for a considerable time the only correction which it was thought neces- 

 sary to make for reduction to a vacuum. But in the year 1828 Bessel, in a very important 

 memoir in which he determined by a new method the length of the seconds' pendulum, pointed 

 out from theoretical considerations the necessity of taking account of the inertia of the air as 

 well as of its buoyancy. The numerical calculation of the effect of the inertia forms a 

 problem of hydrodynamics which Bessel did not attack ; but he concluded from general 

 principles that a fluid, or at any rate a fluid of small density, has no other effect on the 

 time of very small vibrations of a pendulum than that it diminishes its gravity and increases 

 its moment of inertia. In the case of a body of which the dimensions are small compared 

 with the length of the suspending wire, Bessel represented the increase of inertia by that of a 

 mass equal to k times the mass of the fluid displaced, which must be supposed to be added 

 to the inertia of the body itself. This factor k he determined experimentally for a sphere a 

 little more than two inches in diameter, swung in air and in water. The result for air, 

 obtained in a rather indirect way, was k = 0*9459, which value Bessel in a subsequent paper 

 increased to 0'956. A brass sphere of the above size having been swung in water with two 

 different lengths of wire in succession gave two values of k, differing a little from each 

 other, and equal to only about two-thirds of the value obtained for air. 



