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



quently employed in this subject. It consists in supposing an exactly symmetrical motion to 

 take place on the opposite side of the rigid plane, by which means we may evidently conceive 

 the plane removed. 



Let the sphere be oscillating in the direction of the axis of >r, the oscillations in this case, as 

 in the last, beinw so small that they may be taken as rectilinear in calculating the motion of the 

 fluid ; and instead of a rigid plane conceive an equal sphere to exist at an equal distance on the 

 opposite side of the plane xy, moving in the same direction and with the same velocity as the 

 actual sphere. Let r, 9, ai, be the polar co-ordinates of any particle measured from the centre of the 

 sphere, 6 beinw the angle between r and a line drawn through the centre parallel to the axis of -v, 

 and w the ano-le which the plane passing through these lines makes with the plane xz. Let /, ff, w, 

 be the corresponding quantities symmetrically measured from the centre of the imaginary sphere. 



If the fluid were infinite we should have for the motion corresponding to that of the given 

 sphere 



Ca^ cos , , 



0= -Z-T- (20). 



The motion reflected from the plane is evidently the same as that corresponding to the motion 

 of the imaginary sphere in an infinite mass of fluid, for which we have 



CaVose' , ^ 



^-- 2,-. (2')- 



Now / cos 9' = r cos 9, r sin 9' sin w = r sin 9 sin w, r sin 9' cos w + r sin 9 cos to = 2 A ; 

 whence r' = r^ + 4 A' — 4Ar sin 9 cos w, 



and equation (21) is reduced to 



Ca^r COS0 

 ' 2 \r^ + iK- - i hr &\n 9 cos w\^ 



o 

 the above equation is reduced to 



Retainnie only the terms of the order — -- or -r- , so as to get the value of — ^ to the order — , 

 ° '' «■' h dr h 



Ca'r cos9 , , 



•^= I6F- (^^>' 



and the value of — ^ when r = « is, to the required degree of approxim.ation, 

 dr 



Co? cos 9 

 TSA^ ■ 



For the value of rf) corresponding to the motion of the imaginary sphere reflected from the real 

 sphere, we shall therefore have ' 



Co" cos 9 



'^=--iiAV^ ^^^>- 



Adding together the values of <p given by (20), (22) and (23), putting r = u, and replacing 



C bv — , we have, to the requisite degree of approximation, 

 ' dt 



d(b at 3 a'\ dC 



dF^-^VTei?)^-'"'^- 



Hence in this case the motion of the sphere will be the same as if an additional mass equal to 



(1 + -I — were collected at its centre. The eifect therefore of a distant rigid plane which is 

 16 AV 2 ^ ^ 



