Mn. STOKES, ON SOME CASES OF FLUID MOTION. 107 



ferential. For the same reason, in the progressive motion of a solid in a fluid, the effect of friction 

 continually accumulating, the motion might at last be sensibly different from what it would be if 

 there were no friction, and that, even if the friction were very small. In the case of small oscillatory 

 motions however it would appear that the effect of friction in the forward oscillation, supposing that 

 friction small, would be counteracted by its effect in the backward oscillation, at least if the two 

 were symmetrical. In this case then we might expect our results to agree very nearly with experi- 

 ment, so far at least as the time of oscillation is concerned. 



The forces which act on the fluid are supposed in the following investigations to be such that 

 Xdx + Ydy + Zdz is the exact differential of a function of x, y and z, where X, Y, Z, are the 

 components, parallel to the axes, of the accelerating force acting on the particle whose co-ordinates 

 are x, y, z. The only effect of such forces, in the case of a homogeneous, incompressible fluid, being 

 to add the quantity p j(Xdx + Ydy + Zdz) to the pressure, the forces, as well as the pressure due 

 to them, will for the future be omitted for the sake of simplicity. 



2. It is a recognised principle, and one of great importance in these investigations, that when 

 a problem is determinate any solution which satisfies all the requisite conditions, no matter how ob- 

 tained, is the solution of the problem. In the case of fluid motion, when the initial circumstances 

 and the conditions with respect to the boundaries of the fluid are given, the problem is determinate. 

 If it were required to find what sort of steady motion could take place between given surfaces, the 

 problem would not be determinate, since different kinds of steady motion might result from different 

 initial circumstances. 



It may be well here to enumerate the conditions which must be satisfied in the case of a homoge- 

 neous incompressible fluid without a free surface, the case which is considered in this paper. We 

 have first the equations, 



] dp 1 dp 1 dp 



--— =--Zir - —- = - nr-,, -—■ 

 p aw p dy p dz 



du du du du 



putting 73-, for — — h u -J- + f — — ^^;r~^ 3"" "^it "^31 for the corresponding quantities for y 

 at ax ay a z 



and z, and omitting the forces. 



We have also the equation of continuity, 



du dv dw 

 ax ay dz 

 (J) and (jB) hold at all times for all points of the fluid mass. 



If <r be the velocity of the point (.r, y, z) of the surface of a solid in contact with the fluid 

 resolved along the normal, and v the velocity, resolved along the same normal, of the fluid particle, 

 which at the time t is in contact with the above point of the solid, we must have 



V = (J (.a)*-, 



at all times and for all points of the fluid which are in contact with a solid. 



If tlie fluid extend to infinity, and the motion at first be zero at an infinite distance, we must 

 have 



u = 0, V = 0, w = 0, at an infinite distance (6). 



An analagous condition is, that the motion shall not become infinitely great about a particular 

 point, as the origin. 



• for greater cIcarnesB, those equationii whicll mu«t hold for all values of the variables, or of some of them, are denoted by small 

 value» of the variables within limits depcndinK on the problem letters. The latter class serve to deleriiiine the forms of the arbi- 

 •re denoted by capitals, while those which hold only for certain ' trary functions contained in the integrals of the former. 



02 



'l) r-=-'Z3'a, - — = - STi, (A); 



