Mb. stokes, on SOME CASES OF FLUID MOTION. 135 



The numerical calculation of this ratio is very easy, on account of the great rapidity with 



which the series contained in it converge, both on account of the coefficients, and on account of the 



A B C 



rapid diminution of the angles 0„ and Q'„. The values of — and — will be derived from that of — 



by putting c for a in the first case, and c for b in the second. The calculation of the small motions 

 of the box will thus be reduced to a question of ordinary rigid dynamics. 



These results appear capable of being accurately compared with experiment. For this purpose 

 it will only be necessary to attach a box, capable of containing fluid, to a rigid body oscillating 

 as a pendulum. The box may itself form the rigid body. The centre of gravity of the interior 

 of the box should be in a vertical plane passing through the axis of suspension, which will be known 

 by observing whether the position of equilibrium of the whole is affected by filling the box with 

 fluid. The mass, moment of inertia, and depth of the centre of gravity of the solid, including the 

 box, must first be found. The last of these may be found by loading the upper part of the 

 oscillating body till the equilibrium just becomes unstable: the moment of inertia will then be found 

 by means of the time of oscillation when the weight is removed ; or else both may be determined by 

 the times of oscillation when the solid is loaded with another of known mass and form and placed in 

 a known position, and again when it is not loaded. The same must then be done when the box is 

 filled with fluid. We shall thus determine the moment of inertia and depth of the centre of 

 gravity of the fluid; and, subtracting the moment of inertia due to the motion of translation of the 

 fluid, we shall thus get that due to the motion of rotation of the box, and thus determine in 

 succession by observation the quantities J, B and C, or any one of them. These quantities might 

 also be determined by making tlie box oscillate by torsion, and observing the time of oscillation. It 

 must be remembered that the moment of inertia due to the motion of translation of the centre 

 of gravity of the fluid, being capable of being derived from the general dynamical principle, that the 

 motion of the centre of gravity of any system is the same as if the whole mass were collected there, 

 and the external moving forces applied there, is of no use whatever in determining the question 

 of the equality of the pressure in all directions, or that of the amount of friction. It would seem to 

 be most convenient to have the centre of gravity of the fluid in the axis of suspension. In this case 

 if M, M', be the masses of the solid and fluid, /x, /x, their moments of inertia, f, t', the times 

 of oscillation, in seconds, when the box is empty and when it is full respectively, /i the depth of the 

 centre of gravity of the solid, / the length of the second's pendulum, we have 



Ij. + n' = It"^ Mil ; 

 whence fx - l{t'" — t^)Mh. 



If the centre of gravity of the fluid be at a depth h' below the axis of suspension, we shall have 

 fi =l{t"^ — t-) Mh + It'"' M'h' ; in this case /x' — M'/i' will be the moment of inertia due to the 

 motion of rotation of the box. 



When one of the quantities a, b, becomes infinitely great compared with the other, the ratio 



C 



— becomes 1, as will be seen from equation (40). This result might have been expected. When 



a = b tlie value of — is -ISGSSJ. 



The experiment of the box appears capable of great variety as well as accuracy. We may take 

 boxes in which the edges liave various ratios to each other, and may make the same box oscillate in 

 various positions. 



15. Initial motion in a rectangular box, the several points of the surface of which are moved 

 with given velocities, consistent with the condition that the volume of the fluid is not altered. 



