Mb stokes, on SOME CASES OF FLUID MOTION. 121 



centre of the ball when at rest might not be insensible, and in the second place the oscillations 

 would have to be inconveniently small, in order that the value of (j) which has been given might be 

 sufficiently approximate. The effect of a small slit in the upper part of the case, sufficient to allow 

 the wire by which the ball is supported to oscillate, would evidently be insensible, for the conden- 

 sation being insensible in a vertical plane passing through the axis of rotation, since the alteration of 

 pressure in that plane is insensible, the air would not have a tendency alternately to rush in and out 

 at the slit. 



1 0. Effect of a distant rigid plane on the motion of a ball pendulum. 



Although this problem may be more easily solved by an artifice, it may be well to give the direct 

 solution of it by the method mentioned in Article 6. In order to calculate the motion reflected from 

 the plane, it will be necessary to solve the following problem : 



To find the initial motion at any point of a mass of fluid infinitely extended, except where it 

 is bounded by an infinite solid but not rigid plane, the initial motion of each point of the solid 

 plane being given. 



It is evident that motion directed to or from a centre situated in the plane, the velocity being 

 the same in all directions, and varying inversely as the square of the distance from that centre, 

 would satisfy the condition that udx + vdy + wdz is an exact differential, and would give to 

 the particles in contact with the plane a velocity directed along the plane, except just about 

 the centre. Let us see if the required motion can be made up of an infinite number of such 

 motions directed to or from an infinite number of such centres. 



Let x, y, z, be the co-ordinates of any particle of fluid, the plane xy coinciding with the 

 solid plane, and the axis of z being directed into the fluid. Let x', y, be the co-ordinates of 

 any point in the solid plane : then the part of (p corresponding to the motion of the element 

 dx dy of the plane will be 



\//(a-', y')dx'dy' 



\/{a) - x'Y + {y - y'y + x^ ' 



and therefore the complete value of (p will be given by the equation 



-to .« \lr(x',y')dx'dy' 



J-a>J-a V {(•»? - x'y- + {y - y'y + «*} 



The velocity parallel to z at any point = — 



dz 



-a: 



\|/ («', y')zdx' dy 



\ {x - x'Y + (y- y'y + «'}* * 



Now when z vanishes the quantity under the integral signs vanishes, except for values of x' 

 and y indefinitely near to x and y respectively, the function \j/(x', y) being supposed to vanish 

 when .I'' or y' is infinite. Let then x' = x + ^, y'=y + ri, then, |^ and r/^ being as small as 

 we please, the value of the above expression when z = becomes 



- the limit of / / ^ \^, ^ , ^-j-2 — - when z = 0. 



Now if \j/ (x', y) does not alter abruptly between the limits x - ^ and .v + ^ of *', and y - r; 

 and y + ti^ of y, the above expression may be replaced by 



. , ^ 1 ■• ■ <. r^- /"'■ zdPdn 

 -v// («■,?/) X the hunt of / / -- — \ , 



which is = - 2 7r\//(r, y). 

 Vol. VIII. Paiit I. Q 



