AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 301 



Hence P' = iTff + v (u - u), &c. ; and in the equations (8) we n)ay employ the actual velocities 

 u, V, w, since the pressures P, Q, R are independent of any motion of translation and rotation 

 common to the whole fluid. Hence the equations P' = P, &.c. give us 



which three equations with (15) are those which must be satisfied at the surface of a solid, together 

 with the equation Z, = 0. It will be observed that in the case of a free surface the pressures 

 P', Q', R' are given, whereas in the case of the surface of a solid they are known onlv by the 

 solution of the problem. But on the other hand the form of the surface of the solid is given, 

 whereas the form of the free surface is known only by the solution of the problem. 



Dubuat found by experiment that when the mean velocity of water flowing through a pipe is 

 less than about one inch in a second, the water near the inner surface of the pipe is at rest. 

 If these experiments may be trusted, the conditions to be satisfied in the case of small velocities 

 are those which first occurred to me, and which are included in those just given by supposing v =co . 



I have said that when the velocity is not very small the tangential force called into action by 

 the sliding of water over the inner surface of a pipe varies nearly as the square of the velocity. 

 This fact appears to admit of a natural explanation. When a current of water flows past an 

 obstacle, it produces a resistance varying nearly as the square of the velocity. Now even if the 

 inner surface of a pipe is polished we may suppose that little irregularities exist, forming so many 

 obstacles to the current. Each little protuberance will experience a resistance varying nearly as 

 the square of the velocity, from whence there will result a tangential action of the fluid on the 

 surface of the pipe, which will vary nearly as the square of the velocity ; and the same will be true 

 of the equal and opposite reaction of the pipe on the fluid. The tangential force due to this cause 

 will be combined with that by which the fluid close to the pipe is kept at rest when the velocity 

 is sufficiently small. 



There remains to be considered the case of two fluids having a common surface. Let 

 «', v, w', n', S' denote the quantities belonging to the second fluid corresponding to it, &c, 

 belonging to the first. Together with the two equations L = and (15) we shall have in this 

 case the equation derived from (15) by putting u\ v, w for m, t;, ?« ; or, which comes to the 

 same, we shall have the two former equations with 



I (It - u') + m (v - v') + n {zv - w') = (18) 



If we consider the principles on which equations (17) were formed to be applicable to the 

 present case, we shall have six more equations to be satisfied, namely (l7), and the three 

 equations derived from (17) by interchanging the quantities referring to the two fluids, and 

 changing the signs of I, m, n. These equations give the value of sr, and leave five equations 

 of condition. If we must suppose v = cc , as appears most probable, the six equations above 

 mentioned must be replaced by the six u = u, v' = u, w = w, and 



Ip - ixf{u,v, w) ^ Ip' - iu'/('*'i "'i «"')' &c., 

 f(u,v,w) denoting the coefficient of /u in the first of equations (17). We have here six equations 

 of condition instead of five, but then the equation (I8) l)ecoines identical. 



7- The most interesting questions connected with this subject require for their solution a 

 knowledge of the conditions which must be satisfied at the surface of a solid in contact with 

 the fluid, which, except perhaps in case of very small motions, are unknown. It may be 

 well however to give some applications of the preceding equations which an independent of 

 these conditions. Let us then in the first place consider in what manner the transmission of 

 sound in a fluid is affected by the tangential action. To take the simplest case, sup])ose that 

 no forces act on the fluid, so that the pressure and density are constant in the state of 



