[20] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



the envelopes should be similar and similarly situated with respect to the solids oscillating 

 within them, and that their linear dimensions should be in the same ratio as those of the 

 oscillating bodies. In strictness, it is likewise necessary that the solids should be similarly 

 situated with respect to the axis of rotation. If however two similar solids, such as two 

 spheres, are attached to two fine wires, and made to perform small oscillations in two 

 unlimited masses of fluid, and if we agree to neglect the effect of the suspending wires, and 

 likewise the effect of the rotation of the spheres on the motion of the fluid, which last will 

 in truth be exceedingly small, we may regard the two systems as geometrically similar, and 

 they will be dynamically similar provided the condition (6) be satisfied. When the two 

 fluids are of the same nature, as for example when both spheres oscillate in air, the condition 

 of dynamical similarity reduces itself to this, that the times of oscillation shall be as the 

 squares of the diameters of the spheres. 



If, with Bessel, we represent the effect of the inertia of the fluid on the time of oscillation 

 of the sphere by supposing a mass equal to k times that of the fluid displaced added to the 

 mass of the sphere, which increases its inertia without increasing its weight, we must expect 

 to find k dependant on the nature of the fluid, and likewise on the diameter of the sphere. 

 Bessel, in fact, obtained very different values of k for water and for air. Baily's experiments 

 on spheres of different diameters, oscillating once in a second nearly, shew that the value of 

 k increases when the diameter of the sphere decreases. Taking this for the present as the 

 result of experiment, we are led from theory to assert that the value of k increases with the 

 time of oscillation ; in fact, k ought to be as much increased as if we had left the time of 

 oscillation unchanged, and diminished the diameter in the ratio in which the square root of 

 the time is increased. It may readily be shewn that the value of k obtained by Bessel's 

 method, by means of a long and short pendulum, is greater than what belongs to the long 

 pendulum, much more, greater than what belongs to the shorter pendulum, which oscillated 

 once in a second nearly. The value of k given by Bessel is in fact considerably larger than 

 that obtained by Baily, by a direct method, from a sphere of nearly the same size as those 

 employed by Bessel, oscillating once in a second nearly. 



The discussion of the experiments of Baily and Bessel belongs to Part II. of this paper. 

 They are merely briefly noticed here to shew that some results of considerable importance 

 follow readily from the general equations, even without obtaining any solution of them. 



7- Before proceeding to the problems which mainly occupy this paper, it may be well to 

 exhibit the solution of equations (2) and (3) in the extremely simple case of an oscillating plane. 



Conceive a physical plane, which is regarded as infinite, to be situated in an unlimited 

 mass of fluid, and to be performing small oscillations in the direction of a fixed line in the 

 plane. Let a fixed plane coinciding with the moving plane be taken for the plane of yz, the 

 axis of y being parallel to the direction of motion, and consider only the portion of fluid 

 which lies on the positive side of the plane of yz. In the present case, we must evidently 

 have u = 0, w = ; and p, v will be functions of x and t, which have to be determined. The 

 equation (3) is satisfied identically, and we get from (2), putting m = n'p, 



dp dv , d'v 



7- = 0, — = n — j (8) 



dx dt dx* 



