Mr. stokes, on SOME CASES OF FLUID MOTION. 123 



whence [q" + q' — 2qq' cos (w — to) + is''} '■■' = {h" + r' + q'" - 2hr cos 9 — 2q'r sin 9 cos (to — a/) ] "■'. 

 Now supposing the ratio of a to h to be very small, and retaining the most important term, the value 



of ~ when r = a will be equal to the coefficient of r when (p is expanded in a series ascend- 

 ing according to powers of r, 



Ca^ r" r-" {2 K' — q') \h coi 9 +q' sin 9 cos (fi) — w')\q'dq'dw' 



1^3,. ^ r (fih'' -q'^)q'dq' Ca'cos9 , ^ 



= - aC«^Acos0_f (4^H-,'T = - -^^ ^''^- 



In order now to determine the motion reflected from the plane and again from the sphere, 



we must suppose the several points of the sphere to be moved with a normal velocity , 



oh, 

 or, which is the same, we must suppose the whole sphere to be moved towards the plane with 



a velocity — — - . Hence the value of corresponding to this motion will be given by the equation 

 8 h 



Ca^ cos 9 



1'=--uif^ <'«)■ 



For points at a great distance from the centre of the sphere, the motion which is twice 

 reflected will be very small compared with that which is but once reflected. For points close to 

 the sphere however, with which alone we are concerned, those motions will be of the same order 

 of magnitude, and if we take account of the one we must take account of the other. 



Putting q = r sin 9, z = h-rcos9 in (iG), expanding, and retaining the two most important 

 terms, we have 



^ /" „ a'r cos 9\ 



K being a constant, the value of which is not required, and the second term being evidently found 

 by multiplying the quantity at the second side of (17) by r. Adding together the parts of d) 



given by equations (15), (18) and (19), putting r = a, replacing C by -— , and taking for h the 



value which it has in equilibrium, just as in the case of the oscillating cylinder in Article 8, we 

 have for the small motion of the sphere 



dd) dC a I Za\ dC 



_Z = ^ _ _ 1 + -— -- cos 0. 

 dt dt 2 \ Sh'J dt 



The resultant of the part of the pressure due to the first term is zero : that due to the 



second term is greater than if the plane were removed in the ratio of 1 n to 1. Conse- 



a' 

 quently, if we neglect quantities of the order — , the effect of the inertia of the fluid is, to add 



It 



a mass equal to I ^ + ~ 71 ) 7 to that of the sphere, without increasing the moment of inertia 



of the latter about its diameter. The effect therefore of a large spherical case is eight times as 

 great as that of a tangent plane to the case, perpendicular to the direction of the motion of 

 the ball. 



The efl'ect of a distant rigid plane parallel to the direction of motion of an oscillating 

 sphere might be calculated in the same manner, but as the method is sufficiently explained by 

 the first case, it will be well to employ the artifice before alluded to, an artifice whicii is fre- 



