Mr. stokes, on SOME CASES OF FLUID MOTIOX. Ill 



element of the bounding surface, the motion of all particles in contact with those surfaces is 

 tangential. 



A result which follows at once from this principle, and which appears to admit of comparison 

 with experiment, is the following. Conceive an ellipsoid, or any body which is symmetrical with 

 respect to three planes at right angles to each other, to be made to oscillate in a fluid in the 

 direction of each of its three axes in succession, the oscillations being very small. Then, in each 

 case, as may be shown by the same sort of reasoning as that employed in Art. 8, in the case 

 of a cylinder, the effect of the inertia of the fluid will be to increase the mass of the solid by 

 a mass having a certain unknown ratio to that of the fluid displaced. Let the axes of co-ordinates 

 be parallel to the axes of the solid ; let <r, y, n, be the co-ordinates of the centre of the solid, 

 and let M, M\ M" , be the imaginary masses which we must suppose added to that of the solid 

 when it oscillates in the direction of the axes of ,r, y, z, respectively. Let it now be made to oscillate 

 in the direction of a line making angles «, /3, y, with the axes, and let s be measured along 

 this line. Then the motions of the fluid due to the motions of the solid in the direction of the 

 three axes will be superimposed. The motion being supposed to be small, the resultant of the 



pressures of the fluid on the solid will be three forces, equal to M cos a , M' cos & — -, 



Qi o ft c 



M"cosy--~^, respectively, in the directions of the three axes. The resultant of these in the 



d's 



direction of the motion will be M where 



' dt- 



jl/ = M cos V -I- 31' cos ^/3 + M" cos -7. 



Each of the quantities M, M', M' and J/, may be determined by observation, and we may 



And whether the above relation holds between them. Other relations of the same nature may be 



deduced from the principle explained in this article. 



6. Reflection. 



Conceive two solids, A and B, immersed in a fluid of infinite extent, the whole being at rest. 

 Suppose A to be moved in any manner by impulsive forces, while B is held at rest. Suppose 

 the solids A and B of such forms that, if either were removed, and the several points of the 

 surface of the other moved instantaneously in any given manner, the motion of the fluid could 

 be determined : then the actual motion can be approximated to in the following manner. Conceive 

 the place of B to be occupied by fluid, and A to receive its given motion ; then bv hypothesis 

 the initial motion of the fluid can be determined. Let the velocity with which the fluid in 

 contact with that which is su])posed to occupy B's place penetrates into the latter be found, 

 and then suppose that the several points of the surface of B are moved with normal velocities 

 equal and opposite to those just found, jl's place being supposed to be occupied by fluid. The 

 motion of the fluid corresponding to the velocities of the several points of the surface of B can 

 then be found, and A must now be treated as B has been, and so on. The system of velocities 

 of the particles of the fluid corresponding to the first system of velocities of the particles of the 

 surface of B, form what may be called the motion of A rejtected from B; the motion of the 

 fluid arising from the second system of velocities of the particles of the surface of A may be 

 called the motion of A rejlecfed from B mid again from A, and so on. It must be remembered 

 that all these motions take place simultaneously. It is evident that these reflected motions will 

 rapidly decrease, at least if the distance between A and B is considerable compared with their 

 diameters, or rather with the diameter of cither. In this case the calculation of one or two 

 reflections will give the motion of the fluid due to that of A with great accuracy. It is evident 

 that the princi|)le of reflection will extend to any number of solid bodies inuncrsed in a fluid ; 

 or again, the body B may be supposed to be hollow, and to contain the fluid and A, or else 

 A to contain B. In some cases the series arising from the successive reflections can be sununed. 



