104 



December 9, 1850. 



On the effect of the internal friction of Fluids on the Motion of 

 Pendulums. By Professor Stokes. 



It has been acknowledged for some time that the results which 

 follow from the common theory of fluid motion relative to the effect 

 of a fluid on the time of vibration of a pendulum do not agree well 

 with observation. The volume of the Philosophical Transactions for 

 1832 contains the results obtained experimentally by the late Mr. 

 Baily relating to the effect of air in altering the time of vibration of 

 a great variety of pendulums. The experimental results are exhibited 

 by the value of n, the factor by which the correction for buoyancy 

 must be multiplied in order to give the whole effect observed. With 

 pendulums composed of spheres suspended by fine wires, Baily found 

 n~ 1*864 for spheres a little less than \\ inch in diameter, and 

 rc-= 1*748 for spheres about 2 inches in diameter. The result which 

 follows from the common theory is n=l'5, as was first shown by 

 Poisson. The value T864 was the mean of 16 pair of experiments, 

 giving a mean error 0*023, and 1*748 was the mean of 12 pair, which 

 gave a mean error 0*014, so that the difference between the two re- 

 sults, and between either of them and the common theory, is far too 

 large to be attributed to errors of observation. 



The chief object of this paper was, to apply to the calculation of 

 the motion of a pendulum the general equations of motion which are 

 arrived at when the internal friction of the fluid is taken into account, 

 and to compare the resulting formulae with the experiments of Baily 

 and others. The general equations, simplified, first, by neglecting 

 the square of the velocity, secondly, by neglecting the compressibility 

 of the fluid, the effect of which in the present instance is in fact quite 

 insignificant, thirdly, by omitting the external forces, the effect of 

 which may be taken into account separately, are 



1 dp ,(dru d*u d?u\ du ,, x 



i£ = *^+dr + d?)-dt >8iC (L) 



The second and third of the general equations are not written down, 

 because they may be supplied by symmetry. In these equations p is 

 the density, p the mean of the normal pressures in the direction of any 

 three rectangular planes passing through the point of which x, y, z 

 are the coordinates ; u is the velocity in the direction of x, t the time, 

 and /x' a certain constant, depending upon the nature of the fluid, 

 which the author proposes to call the index of friction. 



The author has succeeded in obtaining the solution of equations 

 (1.) in the two cases of a sphere and of an infinite cylinder. The 

 latter may be applied to the case of a pendulum consisting of a long 

 cylindrical rod, by treating each element of the rod as belonging to 

 an infinite cylinder oscillating with the same linear velocity. The 

 following is the solution in the case of a sphere, so far as relates to 

 the resultant action of the fluid on the sphere. 



Let £ be the abscissa of the centre of the sphere, measured in the 



