66 EBPOKT— 1882. 



upon, but the simpler and more important case where the motion begins 

 from rest is worked out in detail. If the displacement of the centre of 

 the sphere be given by the equation £ = c sin nt, then the stream func- 

 tion, when the surrounding fluid is infinite, is 



^L = ^a?c?.m^d\{ ( 1 +-^')cosn< + -^fl + -\innt )- 



- — ( cos Oat - vr + va) + f 1 + — ") sin (...)) e-"-"^! 



where r = v'(?;,/2^), whence the pressure and resultant force on the 

 sphere are easily deduced in simple forms. The latter consists of two 

 terms, whose effects on the motion of a pendulum are different. If r 

 is the time of oscillation, the effect of one term is to produce an apparent 

 increase of inertia equal to ItM! , whilst the other has most effect on the 



TT Ic'W 



amplitude of vibration, the log decrement being -— . -— — —-, , where 



^ ' & ° 2r M + IM.' 



M , M' denote the masses of the sphere, and of the fluid displaced by 



it, and 



He then passes on to investigate the effect of a concentric spherical 

 boundary. When the effect of the boundaiy is small it is sufiicient to 

 treat it as absent, and then add small corrections to the results. If the 

 viscosity is small, or if the time of oscillation is small, this correction is 

 the same as if there were no viscosity.' 



We have seen (p. 27) that a steady motion of the fluid when a 

 cylinder moves uniformly through it is impossible. This is not so with 

 the sphere. In this case Stokes shows, that, if V be the velocity of the 

 sphere, ip = :j aW(Srja — ajr) sin ^6 (axes fixed in the sphere), and 

 that the force necessary to maintain the motion is G-rrf^iaV. This varies 

 only as the radius of the sphere, ' accordingly, fine powders remain 

 nearly suspended in a fluid of widely different specific gravity.' If a 

 sphere of density o- be descending through a fluid under gravity, the 

 limiting velocity (if it is not very great) is 2g(t7 — p)a^/9/u'; this for a 

 globule of water in air, of "001 inch diameter, is 1"593 inch per second, 

 whilst for one with a diameter of "0001 inch, it is less than one-sixteenth 

 of an inch per second. 



In a note at the end of this paper it is shown that if a sphere rotate 

 about a diameter the particles of the fluid move, when the motion is 

 steady, in annuli, with velocities given hjv = a'w sin 6 jr^, w being the 

 angular velocity of the sphere ; and that, in general, the motion is given 

 by V = v' sin d, where v' is a function of r alone. This steady motion in 

 annuli is only possible when the motion is slow ; if it is not so, then with 

 the annular motion is combined one in planes through the axis : ' In 

 fact it is easy to see that, from the excess of centrifugal force in the 

 neighbourhood of the equator of the revolving sphere, the particles in 

 that part will recede from the sphere and approach it again in the 

 neighbourhood of the poles, and this circulating motion will be com- 



• See above, p. 61. 



