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



41. It appears then that by supposing the rate of alteration of the velocity of the sphere 

 to decrease indefinitely, we obtain from the solution of the problem of Section II. the same 

 result as was obtained in Art. 39, by treating the motion as steady. As yet, however, the 

 method of Art. 40 is subject to a limitation from which that of Art. 39 is free. In obtaining 

 equation (40), it was supposed that the maximum excursion of the sphere was small in com- 

 parison with its radius. Retaining this restriction while we suppose t to become very large, 

 we are obliged to suppose c to become very small, so that the velocity of the sphere is not 

 merely so small that we may neglect terms depending upon its square, a restriction to which 

 Art. 39 is also subject, but so extremely small that the space passed over by the sphere in even 

 a long time is small in comparison with its radius. 



We have seen, however, that on supposing t very large in (40) we obtain a result 



identical with (124), not merely a result with which (124) becomes identical when the restriction 



above mentioned is introduced. This leads to the supposition that the solution expressed by 



(40) is in fact more general than would appear from the way in which it was obtained. That 



such is really the case may be shewn by a slight modification of the analysis. Instead of 



referring the fluid to axes fixed in space, refer it to axes originating at the centre of the sphere, 



and moveable with it. In the general equations of motion, the terms which contain differential 



coefficients taken with respect to the co-ordinates will remain unchanged, inasmuch as they 



represent the very same limiting ratios as before : it is only those in which differentiation with 



d' 

 respect to t occurs that will be altered. If — be the symbol of differentiation with respect to t 



at 



when the fluid is referred to the moveable axes, we shall have 



d d! d}~ d 

 dt dt dt dx^ 



dP d 

 but the terms arising from -— — are of the order of the square of the velocity, and are 



dt dx 



therefore to be neglected. Hence the general equations have the same form whether the fluid 

 be referred to the fixed or moveable axes. But on the latter supposition the equations of 

 condition (30) become rigorously exact. Hence equation (40) gives correctly the solution of 

 the problem, independently of the restriction that the maximum excursion of the sphere be small 

 compared with its radius, provided we suppose the polar co-ordinates r, 9 measured from the 

 centre of the sphere in its actual, not its mean position. Similar remarks apply to the problem 

 of the cylinder. Moreover, in the case of a sphere oscillating within a concentric spherical 

 envelope, it is not necessary, in order to employ the solution obtained in Section II., that the 

 maximum excursion of the sphere be small compared with its radius ; it is sufficient that it be 

 small compared with the radius of the envelope. 



These are points of great importance, because the excursions of an oscillating sphere in a 

 pendulum experiment are not by any means extremely small compared with the radius of the 

 sphere ; and in the case of a narrow cylinder, such as the suspending wire, so far from 

 the maximum excursion being small compared with the radius of the cylinder, it is, on the con- 

 trary, the radius which is small compared with the maximum excursion. 



