Mr. stokes, on SOME CASES OF FLUID MOTION. 127 



second case as it is for the corresponding point in the first. Also, cacli element of the surface 

 of the second solid will be m" times as great as the corresponding element of the surface of the 

 first. Hence the whole resistance on the second solid will be m? times as great as that on the 

 first, and therefore the quantity n depends only on the form, and not on the size of the solid. 



When forces act on the fluid, it will only be necessary to add the corresponding pressure. 

 Hence when a sphere descends from rest in a fluid by the action of gravity, the motion will be 

 the same as if a moving force equal to that of the sphere mimis that of the fluid displaced 

 acted on a mass equal to that of the sphere plus half that of the fluid displaced. For a cylinder 

 which is so long that we may suppose the length infinite, descending horizontally, every thing 

 will be the same, except that the mass to be moved will be equal to that of the cylinder plus 

 the whole of the fluid displaced. In these cases, as well as in that of any solid which is sym- 

 metrical with respect to two vertical planes at right angles to each other, the motion will be 

 uniformly accelerated, and similar solids of the same material will descend with equal velocities. 

 These results are utterly opposed even to the commonest observation, which shews that large 

 solids descend much more rapidly than small ones of the same shape and material, and that the 

 velocity of a body falling in a fluid, (such as water), does not sensibly increase after a little 

 time. It becomes then of importance in the theory of resistances to inquire what may be the 

 cause of this discrepancy between theory and observation. The following are the only ways of 

 accounting for it which suggest themselves to me. 



First. It has been supposed that the same particles remain in contact with the solid through- 

 out the motion. It must be remembered that we suppose the ultimate molecules of fluids, (if such 

 exist), to be so close that their distance is quite insensible, a supposition of the truth of which 

 there can be hardly any doubt. Consequently we reason on a fluid as if it were infinitely divisible. 

 Now if the motion which takes place in the cases of the sphere and cylinder be examined, sup- 

 posing for simplicity their motions to be rectilinear, it will be found that a particle in contact 

 with the surface of either moves along that surface with a velocity which at last becomes in- 

 finitely small, and that it does not reach the end of the sphere or cylinder from which the whole 

 is moving until after an infinite time, while any particle not in contact with the surface is at 

 last left behind. It seems difficult to conceive of what other kind the motion can be, without 

 supposing a line, (or rather surface) of particles to make an abrupt turn. If it should be said 

 that the particles may come off" in tangents, it must be remembered that this sort of motion is 

 included in the condition which has been assumed with respect to the surface. 



Secondly. The discrepancy alluded to might be supposed to arise from the friction of the 

 fluid against the surface of the solid. But, for the reason mentioned in the beginning of this 

 paper, this explanation does not appear to me satisfactory. 



Thirdly. It appears to me very probable that the spreading out motion of the fluid, whicli 

 is supposed to take place behind the middle of the sphere or cylinder, though dynamically possible, 

 nay, the only motion dynamically possible when the conditions which have been supposed are 

 accurately satisfied, is unstable ; so that the slightest cause produces a disturbance in the fluid, 

 which accumulates as the solid moves on, till the motion is quite changed. Common observation 

 seems to shew that, when a solid moves rapidly througli a fluid at some distance below the 

 surface, it leaves behind it a succession of eddies in the fluid. When the solid has attained its 

 terminal velocity, the product of the resistance, or rather the mean resistance, and any space 

 through which the solid moves, will be equal to half the vis viva of the corresponding j)ortion 

 of its tail of eddies, so tliat the resistance will be measured by the vis viva in tlie lengtli of two 

 units of that tail. So far therefore as the resistance which a ship experiences depends on tlie 

 disturbance of the water, which is independent of its elevation or depression, that shi]) which 

 leaves the least wake ought, according to this view, to be cceteris paribus the best sailer. The 

 resistance on a ship difl'crs from that on a solid in motion immersed in a fluid in the circumstance, 

 that part of tlie resistance is employed in producing a wave. 



