Mr. stokes, on SOME CASES OF FLUID MOTION. 119 



wise, if all the fluid beyond a certain distance from the part disturbed were at first at rest, the 

 velocity at a great distance will ultimately be directed to or from the disturbed part, and will 



be the same in all directions, and will vary as -^ . The coefficient of — will be proportional 



ing on the term in cb. Since the general form of P. is 



1 



? 



to the rate of gain or loss of volume of the part disturbed. If however this rate should be zero, 

 then the most important part of the velocity at a great distance will in general be that depend- 



A cos 9 + B smd cos w + C sin 6 sin w, 

 we easily find, by making use of rectangular co-ordinates, changing the direction of the axes, 



D cos 



and then again adopting polar co-ordinates, that the above term in (p takes the form = — i 



r 



di being measured from same line passing through the origin. The motion will therefore be the 



same as that round a ball pendulum in an incompressible fluid, the centre of the ball being 



in the origin ; a case of motion which will be considered immediately. In order to represent the 



motion at different times, we must suppose the velocity and direction of motion of the ball to 



change with the time. 



The value of (p given by equation (13) is applicable to the determination of the motion of 



a ball pendulum enclosed in a spherical case which is concentric with the ball in its position of 



equilibrium. If C be the velocity of the centre of the ball at the instant when the centres of 



the ball and case coincide, and if 6 be measured from the direction in which it is movino-, we 



o' 



shall have 



/(e) = c'cosa, F(e) = o; 



.-. Po = 0, P, = C cos e, P, = 0, &c., P'o = 0, &c., 

 and the value of (p for this instant is accurately 



~ W - a? V 





which, when & = eo , becomes 



_ Ca^ cos e 



I? ' 



which is the known expression for the value of <p for a sphere oscillating in an infinitely extended, 

 incompressible fluid. 



It may be shewn, by precisely the same reasoning as was employed in the case of the cylin- 

 der, that in calculating the small oscillations of the sphere the value of - - to be employed is 



a' 



dt ( h'\ 



- rr. r. a + -— COS ; 



6' - a' V 2 a'j 



and from the equation p = — p -J- , we easily find that the whole resultant pressure on the 

 sphere in the tlirection of its centre, and tending to retard it is 



3 P-d " " ^ 



and that perpendicular to this direction is zero. Since is the effective force of the centre 



dt 



¥ \ dC 



;ro. Smce — - 

 di 



in the direction of the motion, and that perpendicular to this direction is of the second order. 



