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



Putting for m its value y(l + */ - l), and denoting by M the mass of the fluid displaced by 

 the sphere, which is equal to ^irpa 3 , we get 



IV 4v«/ ^va V val ) 



V 2 ±va) dt 2 4,va \ vaj dt K } 



F=-Mcn 1 + -LK/-1 + JL 1+- H«Vri»| 

 IV 4v«/ ^va V v«/J 



whence 



Since v^ _ 1 nas ^ een eliminated, this equation will remain unchanged when we pass from the 

 symbolical to the real values of F and £. 



Let t be the time of oscillation from rest to rest, so that nr = ir, and put for shortness 

 k, k' for the coefficients of M' in (51); then 



2/ot t 4j/a 4i>a V val 



(52) 



The first term in the expression for the force F has the same effect as increasing the inertia 



of the sphere. To take account of this term, it will be sufficient to conceive a mass k M' 



collected at the centre of the sphere, adding to its inertia without adding to its weight. The 



main effect of the second term is to produce a diminution in the arc of oscillation : its effect 



on the time of oscillation would usually be quite insensible, and must in fact be neglected 



for consistency's sake, because the motion of the fluid was determined by supposing the motion 



of the sphere permanent, which is only allowable when we neglect the square of the rate of 



decrease of the arc of oscillation. 



If we form the equation of motion of the sphere, introducing the force F, and then 



proceed to integrate the equation, we shall obtain in the integral an exponential e~ 8 ' multi- 



df rp£ 



plying the circular function, S being half the coefficient of •— divided by that of — | . Let 

 1 * ° dt dt 2 



M be the mass of the sphere, My 1 its moment of inertia about the axis of suspension, then 



nk'M 1 (l + ay = <2§ {My* +kM'(l+ af) . 



In considering the diminution of the arc of oscillation, we may put I + a for y. During i 

 oscillations, let the arc of oscillation be diminished in the ratio of J to A t , then 



A . , -n-i k'M 1 



\og>.— =iTd= — — — — , (53) 



& 'A t 2 M+kM v ' 



For a given fluid and a given time of oscillation, both k and k' increase as a decreases. 

 Hence it follows from theory, that the smaller be the sphere, its density being supposed given, 

 the more the time of oscillation is affected, and the more rapidly the arc of oscillation 

 diminishes, the alteration in the rate of diminution of the arc due to an alteration in the radius 

 of the sphere being more conspicuous than the alteration in the time of oscillation. 



21. Let us now suppose the fluid confined in a spherical envelope. In this case, we have 



