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



Resuming the value of (p given by equation (6), let us suppose that the interior cylindrical 

 surface is riirid, and moved with a velocity C in the direction from which is measured, the 

 outer surface being at rest: then f{6) = Ccos9, F(6)=0; whence C, = C, and the other co- 

 efficients are each zero. We have then 



1'=-^^-[-^^l'°'^ (')■ 



Suppose now that the inner cylinder has a small oscillatory motion about an axis parallel 

 to the axes of the cylinders, the cylinders having their axes coincident in the position of 

 equiHbrium. Let x^ be the angle which a plane drawn through the axis of rotation, and that of the 

 solid cylinder at any time makes with a vertical plane drawn through the former. The motion of 

 translation of the axis of the cylinder will differ from a rectilinear motion by quantities depending 

 on \l/-: the motion of rotation about its axis will be of the order \|/, but will have no effect on 

 the fluid. Therefore in considering the motion of the fluid we may, if we neglect squares of yp, 

 consider the motion of the cylinder rectilinear. The expression given for cp by equation (8) will 

 be accurately true only for the instant when the axes of the cylinders coincide; but since the 

 whole resultant pressure on the solid cylinder in consequence of the motion is of the order \p, 

 we may, if we neglect higher powers of \j/ than the first, employ the approximate value of cp 

 given by equation (8). Neglecting the square of the velocity, we have 



d(b 



dd) 

 In finding the complete value of — ^ it would be necessary to express (p by co-ordinates re- 

 ferred to axes fixed in space, which after differentiation we might suppose to coincide with others 

 fixed in the body. But the additional terms so introduced depending on the square of the velocity, 

 which by hypothesis is neglected, we may differentiate the value of (p given by equation (8) as if 

 the axes were fixed in space. We have then, to the first order of approximation, 



„rfC 



d(p dt ib-- 1 



dt b--a' { r J 



If I be the length of the cylinder, the pressure on the element ladO, resolved parallel to .v 

 and reckoned positive when it acts in the direction of .r, 



cos^edO; 



dt (P ] 

 - a' [a J 



b- 



and integrating from = to 9 = 2-ir, we have the whole resultant pressure parallel to ,r 



b' + a- dC 



b' -a' ^ dt 



Since —— is the effective force of the axis, parallel to x, and that parallel to y is of the order >^^ 



we see that the effect of the inertia of the fluid is to increase the mass of the cylinder by 



— n, where fx is the mass of the fluid displaced. This imaginary additional mass must be 



supposed to be collected at the axis of the cylinder. 



If the cylinder oscillate in an infinitely extended fluid 6 = co, and the additional mass becomes 

 equal to that of the fluid displaced. This appears to be a result capable of being compared with 

 experiment, though not with very great accuracy. Two cylinders of the same material, and of the 



