XXX. Supplement to a Memoir On some Cases of Fluid Motion. By 

 George G. Stokes, M.A., Fellow of Pembroke College. 



[Read Nov. 3, 1846.] 



In a memoir which the Society did me the honour to publish in their Transactions*, I showed that 

 when a box whose interior is of the form of a rectangular parallelepiped is filled with fluid and made 

 to perform small oscillations the motion of the box will be the same as if the fluid were replaced by a 

 solid having the same mass, centre of gravity, and principal axes as the solidified fluid, but different 

 moments of inertia about those axes. The box is supposed to be closed on all sides, and it is also 

 supposed that the box itself and the fluid within it were both at rest at the beginning of the motion. 

 The investigation was founded upon the ordinary equations of Hydrodynamics, which depend upon 

 the hypothesis of the absence of any tangential force exerted between two adjacent portions of a fluid 

 in motion, an hypothesis which entails as a necessary consequence the equality of pressure in all 

 directions. The particular case of motion under consideration appears to be of some importance, 

 because it affords an accurate means of comparing with experiment the common theory of fluid 

 motion, which depends upon the hypothesis just mentioned. In my former paper, I gave a series 

 by means of which the numerical values of the principal moments of the solid which may be substi- 

 tuted for the fluid might be calculated with facility. The present supplement contains a different 

 series for the same purpose, which is more easy of numerical calculation than the former. The com- 

 parison of the two scries may also be of some interest in an analytical point of view, since they appear 

 under very different forms. I have taken the present opportunity of mentioning the results of some 

 experiments which I have performed on the oscillations of a box, such as that under consideration. 

 The experiments were not performed with sufficient accuracy to entitle them to be described in 

 detail. 



The calculation of the motion of fluid in a rectangular box is given in the 13th article of my 

 former paper. I shall not however in the first instance restrict myself to a rectangular parallelepiped, 

 since the simplification which I am about to give applies more generally. Suppose then the problem 

 to be solved to be the following. A vessel whose interior surface is composed of any cylindrical 

 surface and of two planes perpendicular to the generating lines of the cylinder is filled with a homo- 

 geneous, incompressible fluid ; the vessel and the fluid within it having been at first at rest, the 

 former is then moved in any manner ; required to determine the motion of the fluid at any instant, 

 supposing that at that instant the vessel has no motion of rotation about an axis parallel to the gene- 

 rating lines of the cylinder. 



I shall adopt the notation of my former paper, ti, v, w are the resolved parts of the velocity at 

 any point along the rectangular axes of x, y, z. Since the motion begins from rest we shall have 

 udx + i^dy + wdz an exact differential d(p. Let the rectangular axes to which the fluid is referred 



• Vol. VIII., Pari I., p. lU.'i. 



Vol.. VIII. Pakt III. AG 



