122 Mr. stokes, ON SOME CASES OF FLUID MOTION. 



If now f(a:\ y) be the given normal velocity of any point {x , y) of the solid plane, the ex- 

 pression for (p given by equation (14) may bf made to give the required normal velocity of 

 the fluid particles in contact with the solid plane by assuming 



^(M,y') = ~— f {'<'', y'), 



whence 



_ _1_ r" /•" /(.r, y')d.v'dy' 



^ 2^7— ./-» {(a? - w'Y + {y- y'f + «=} J " 



This expression will be true for any point at a finite distance from the plane xy even when f{x', y) 

 does alter abruptly ; for we may first suppose it to alter continuously, but rapidly, and may then 

 suppose the rapidity of alteration indefinitely increased : this will not cause the value of (p just 

 given to become illusory for points situated without the plane xy. 



If it be convenient to use polar co-ordinates in the plane xy, putting x = q cos to, y = q sin w, 

 x' = q cos o)', y = q sin &>', and replacing fix , y) by /(?', u>), the equation just given becomes 



, ^ _ J_ r" r-" f{q\ w')q'dq'dw' 



^ Stt^o •'o 5</' + 9" - 297' cos (w -«)') + «=}i' 



To apply this to the case of a sphere oscillating in a fluid perpendicularly to a fixed rigid 

 plane, let a be the radius of the sphere, and let its centre be moving towards the plane with 

 a velocity C at the time t. Then, (Art. 4), we may calculate the motion as if it were produced 

 directly by impact. Let h be the distance of the centre of the sphere from the fixed plane 

 at the time t, and let the line h be taken for the axis of x, and let r, 6, be the polar co-or- 

 dinates of any point of the fluid, r being the distance from the centre of the sphere, and 9 the 

 angle between the lines r and h. Then if the fluid were infinitely extended around the sphere 

 we should have 



Ca" COS0 

 d)= ^^ — (15). 



The velocity of any particle, resolved in a direction towards the plane, =— -Z-cos^ ^ sin (J 



dr rdO 



= ^We-lsin=ei. 



For a particle in the plane xy we have 



r cos Q = h, r sin 6 = q', 

 and the above velocity becomes 



Ca'^Zh'-q'') 



Zih' + q')^ 

 We must now, according to the method explained in (Art. 6), suppose the several points of 

 the plane xy moved with the above velocity parallel to x. We have then 



j(g>w) = —5- ; 



2(h' + q'-y 



whence, for the motion of the sphere reflected from the plane, 



^_Ca^ r- r^' (%h'-q')q'dqdj ^^^^ 



*T A \ {h' + q'^Y\q^ + q'-Zqq' COS {w-w) + x''\''' 



We must next find the velocity, corresponding to this value of cb, with which the fluid pene- 

 trates tlie surface of the sphere. We have in general 



z = h - r cos 0, q = r sin 0, 



