298 Mil. STOKES, ON THE FRICTION OF FLUIDS IN MOTION, 



These equations are applicable to the determination of the motion of water in pipes and canals, 

 to the calculation of the effect of friction on the motions of tides and waves, and such questions. 



If the motion is very small, so that we may neglect the square of the velocity, we may 



put — = ^ , &c. in equations (13). The equations thus simplified are applicable to the 



Dt dt 

 determination of the motion of a pendulum oscillating in water, or of that of a vessel filled with 

 water and made to oscillate. They are also applicable to the determination of the motion of 

 a pendulum oscillating in air, for in this case we may, with hardly any error, neglect the 

 compressibility of the air. 



The case of the small vibrations by which sound is propagated in a fluid, whether a 



liquid or a gas, is another in which — - may be neglected. For in the case of a liquid reasons 



dp 



have been shown for supposing /x to be in (/e pendent of p, and in the case of a gas we may neglect 



— , if we neglect the small change in the value of fj, arising from the small variation of 

 dp 



pressure due to the forces X, Y, Z. 



6. Besides the equations which must hold good at any point in the interior of the mass, 

 it will be necessary to form also the equations which must be satisfied at its boundaries. Let 

 J/ be a point in the boundary of the fluid. Let a normal to the surface at M, drawn on the 

 outside of the fluid, make with the axes angles whose cosines are /, m, n. Let /*", Q', R' be 

 the components of the pressure of the fluid about M on the solid or fluid with which it is in 

 contact, these quantities being reckoned positive when the fluid considered presses the solid or fluid 

 outside it in the positive directions of x, y, x, supposing I, m and n positive. Let S be a 

 very small element of the surface about M, which will be ultimately plane, S' a plane parallel 

 and equal to S, and directly opposite to it, taken within the fluid. Let the distance between S 

 and S' be supposed to vanish in the limit compared with the breadth of S, a supposition which 

 may be made if we neglect the eff"ect of the curvature of the surface at M ; and let us consider the 

 forces acting on the element of fluid comprised between S and S., and the motion of this 

 element. If we suppose equations (8) to hold good to within an insensible distance from the 

 surface of the fluid, we shall evidently have forces ultimately equal to PS, QS, RS, (P,Q and R 

 being given by equations (9),) acting on the inner side of the element in the positive directions of 

 the axes, and forces ultimately equal to P" S, Q' S, R' S acting on the outer side in the negative 

 directions. The moving forces arising from the external forces acting on the element, and the 

 eft'ective moving forces will vanish in the limit compared with the forces PS, &c. : the same 

 will be true of the pressures acting about the edge of the element, if we neglect capillary 

 attraction, and all forces of the same nature. Hence, taking the limit, we shall have 



P' = i', Q'= Q, R' = R. 



The method of proceeding will be different according as the bounding surface considered is a 

 free surface, the surface of a solid, on the surface of separation of two fluids, and it will be 

 necessary to consider these cases separately. Of course the surface of a liquid exposed to the 

 air is really the surface of separation of two fluids, but it may in many cases be regarded as 

 a free surface if we neglect the inertia of the air: it may always be so regarded if we neglect 

 the friction of the air as well as its inertia. 



Let us first take the case of a free surface exposed to a pressure 11, which is supposed to 

 be the same at all points, but may vary with the time; and let Z, = be the equation to the 

 surface. In this case we shall have P' = 111, Q' = mY\, R' = nil; and putting for P, Q, R their 

 values given by (9), and for P^ &c. their values given by (8), and observing that in this case 

 3 = 0, we shall have 



