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



called into play by the continuous sliding of the fluid over itself was no more than counter- 

 acted by the abrupt sliding of the fluid over the solid. As this appears exceedingly improba- 

 ble a priori, it seems reasonable in the first instance to examine the consequences of supposing 

 that no such abrupt sliding takes place, more especially as the mathematical difficulties of the 

 problem will thus be materially diminished. I shall assume, therefore, as the conditions to be 

 satisfied at the boundaries of the fluid, that the velocity of a fluid particle shall be the same, 

 both in magnitude and direction, as that of the solid particle with which it is in contact. The 

 agreement of the results thus obtained with observation will presently appear to be highly 

 satisfactory. When the fluid, instead of being confined within a rigid envelope, extends indefi- 

 nitely around the oscillating body, we must introduce into the solution the condition that 

 the motion shall vanish at an infinite distance, which takes the place of the condition to be 

 satisfied at the surface of the envelope. 



To complete the determination of the arbitrary functions which would be contained in the 

 integrals of (2) and (3), it would be requisite to put t = in the general expressions for u, v t 

 w, obtained by integrating those equations, and equate the results to the initial velocities sup- 

 posed to be given. But it would be introducing a most needless degree of complexity into the 

 solution to take account of the initial circumstances, nor is it at all necessary to do so for the 

 sake of comparison of theory with experiment. For in a pendulum experiment the pendulum 

 is set swinging and then left to itself, and the first observation is not taken till several oscilla- 

 tions have been completed, during which any irregularities attending the initial motion would 

 have had time to subside. It will be quite sufficient to regard the motion as already going on, 

 and limit the calculation to the determination of the simultaneous periodic movements of the 

 pendulum and the surrounding fluid. The arc of oscillation will go on slowly decreasing, but 

 it will be so nearly constant for several successive oscillations that it may be regarded as 

 strictly such in calculating the motion of the fluid ; and having thus determined the resultant 

 action of the fluid on the solid we may employ the result in calculating the decrement of the 

 arc of oscillation, as well as in calculating the time of oscillation. Thus the assumption of 

 periodic functions of the time in the expressions for u, v, w will take the place of the determi- 

 nation of certain arbitrary functions by means of the initial circumstances, 



4. Imagine a plane drawn perpendicular to the axis of x through the point in the 

 fluid whose co-ordinates are x, y, ss. Let the oblique pressure in the direction of this plane 

 be decomposed into three pressures, a normal pressure, which will be in the direction of x, and 

 two tangential pressures in the directions of y, %, respectively. Let P Y be the normal pressure, 

 and T 3 the tangential pressure in the direction of y, which will be equal to the component in 

 the direction of w of the oblique pressure on a plane drawn perpendicular to the axis of y. 

 Then by the formulae (7), (8) of my former paper, and (3) of the present, 



~ du 



Pt-p-Zv — , (4) 



ax 



_ (du dv\ , ■ 



\dy dxj 



These formulae will be required in finding the resultant force of the fluid on the pendulum, after 



