Mn. STOKES, ON SOME CASES OF FLUID MOTION. 125 



parallel to the direction of the motion of a ball pendulum will be half that of a plane at the 

 same distance, and perpendicular to that direction. It would seem from Poisson's words at page 

 562 of the eleventh volume of the Memoires de VInstitut, that he supposed the effect in the 

 former case to depend on a higher order of small quantities than that in the latter. 



If the ball oscillate in a direction inclined to the plane, the motion may be easily deduced 

 from that in the two cases just given, by means of the principle of superposition. 



11. The values of ^ which have been given for the motion of translation of a sphere and 

 cylinder, do not require us to suppose that either the velocity, or the distance to which tlie 

 centre of the sphere or axis of the cylinder has been moved is small, provided the same particles 

 remain in contact with the surface. The same indeed is true of the values corresponding to a 

 motion of translation combined with a motion of contraction or expansion which is the same in 

 all directions, but varies in any manner with the time. The value of ih corresponding to a motion 



of translation of the cylinder is — , C being the velocity of the axis, and 6 being 



measured from a line drawn in the direction of its motion. The whole resultant of the part of 

 the pressure due to the square of the velocity is zero, since the velocity at the point whose co- 

 ordinates are r, 6, is the same as that at the point whose co-ordinates are r and ir — d. To find the 



resultant of the part depending on -—■ , it will be necessary to express (p by means of co-ordinates 



referred to axes fixed in space. Let Ox, Oy, be rectangular axes passing through the centre of 

 any section of the cylinder, •ar the angle which the direction of motion of the axis makes with 

 Ow, ff the inclination of any radius vector to Ox; then 



<p = — (r cos 9 cos -ST + r sin 9 sin 'sr) 



d' {C'oc + C"y) 

 X- + y" 

 putting C' and C" for the resolved parts of the velocity C along the axes of x and y respectively. 

 Taking now axes Ax', Ay, parallel to the former and fixed in space, putting a and /3 for the 



co-ordinates of O, differentiating (p with respect to t, and replacing — by C', and — by C", 



do at 



and then supposing a and (i to vanish, we have 



/ dC_ dC\ 



d(p a'C 2a'(C'x + C"yy- " V dt ^ ^ dt j 



dt .1'^ + y" (3? + yy 



+ r 



The resultant of the part of the pressure due to the first two terms is zero, since the pressure 



at the point {.v, y) depending on these terms is the same as that at the point (- x, — y). It 



will be easily found that the resultant of the whole pressure parallel to x, and acting in tiie 



d C* 

 negative direction, on a length I of the cylinder, is equal to Trpia- —— , and that parallel to 7 



dt 



,dC" 

 equal to trplar — — . The resultant of these two will be irpta^F, where F is the effective force 



of a point in the axis of the cylinder, and will act in a direction opposite to that of F. Hence 

 the only effect of the motion of the fluid will be, to increase tlie mass of the cylinder by that of 

 the fluid displaced. In a similar manner it may be proved that, when a solid sphere moves in 

 any manner in an infinite fluid, the only effect of the motion of the fluid is to increase the mass 

 of the sphere by half that of the fluid displaced. 



