Mr. stokes, on SOME CASES OF FLUID MOTION. 131 



If the cylinder revolve round its axis with an angular velocity w, the normal velocity of 



the surface at any point will be 2wec sin 20. Since e^ is neglected, we may suppose this normal 



velocity to take place on the surface of a circular cylinder whose radius is c; whence, (Art. 8), the 



corresponding value of (p will be 



toec' . 



sin 29. 



r 



If we suppose all these motions to take place together, we have only, (Art. 5), to add together 

 the values of <p corresponding to each. If we suppose the motion very small, so as to neglect the 



/. , , ■ J , -1. , ,• do) dC , dC . , 



square oi the velocity, we need only retain the terms depending on -— , — — and -— — , in the 



value of — t- , and we may calculate the pressure due to each separately. The resultant of the 

 dt 



pressure due to the term — — will evidently be zero, on account of the symmetry of the corre- 

 sponding motion, while the resultant couple will be of the order e°, since the pressure on any 

 point of the surface, and the perpendicular from the centre on the normal at that point, are each of 



the order e. The pressure due to the term — — will evidently have a resultant in the direction 



dt 



of the major axis of a section of the cylinder ; and it will be easily proved that the resultant 

 pressure on a length I of the cylinder is 7rpc^l(l — 2e) —— . That due to the term will be 



Trpc'l(\ + 2e) -T— , acting along the minor axis. If the cylinder be constrained to oscillate so that 



its axis oscillates in a direction making an angle a with the major axis, and if C" be its velocity, 



which is supposed to be very small, the resultant pressures along the major and minor axes will be 



dC" . dC" 



IX {l — 2e) cos a — — and /i (l + 2e) sin a — — respectively, where fx is the mass of the fluid displaced. 



dC" 

 Resolving these pressures in the direction of the motion, the resolved part will be ^ (1 - 2 e cos 2 a) , 



g2 rf C" 



or /u (1 cos 2a) — — , e being the eccentricity ; so that the effect of the inertia of the fluid will be, 



to increase the mass of the solid by a mass equal to /u (1 cos 2a), which must be supposed to be 



collected at the axis. 



A similar method of calculation would apply to any given solid differing little either from 

 a circular cylinder or from a sphere. In the latter case it would be necessary to use expansions 

 in series of Laplace's coefficients, instead of expansions in series of sines and cosines. 



13. Motion of Jluid in a closed bow whose interior is of the form of a rectangular parallelopiped. 



The motion being supposed to begin from rest, the motion at any time may be supposed 

 to have been produced by impact (Art. 4). The motion of the box at any instant may be 

 resolved into a motion of translation and three motions of rotation about three axes parallel to 

 the edges, and passing through the centre of gravity of the fluid, and the part of <f> due to 

 each of these motions may be calculated separately. Considering any one of the motions of 

 rotation, we shall sec that the normal velocity of each face in consequence of it will ultimately 

 be the same as if that face revolved round an axis passing tlirough its centre, and that tile 

 latter motion would not alter the volume of the fluid. Consequently, in calculating tlie part of 



r2 



