AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 303 



The velocity of sound in air, deduced from the note given by a known tube, is sensibly 

 less than that determined by direct observation. Poisson thought that this might be due to the 

 retardation of the air by friction against the sides of the tube. But from the above investigation 

 it seems unlikely that the effect produced by that cause would be sensible. 



The equation (21) may be considered as expressing in all cases the effect of friction; for 

 we may represent an arbitrary disturbance of the medium as the aggregate of series of plane 

 waves propagated in all directions. 



8. Let us now consider the motion of a mass of uniform inelastic fluid comprised 

 between two cylinders having a common axis, the cylinders revolving uniformly about their 

 axis, and the fluid being supposed to have attained its permanent state of motion. Let the 

 axis of the cylinders be taken for that of z, and let q be the actual velocity of any particle, 

 so that u = - q smQ, v = q cos Q, w = 0, r and being polar co-ordinates in a plane parallel to xy. 



d'f d-f d'f 1 df 1 d'/ 

 Observing that ^ + — =_ + -— + -— , where / is any function of a: and y, and 



dp 

 that -— = 0, we have from equations (13), supposing after differentiation that the axis of .t 

 d6 



coincides with the radius vector of the point considered, and omitting the forces, and the part 

 of the pressure due to them, 



dp q 



^ - = 0, 



dr r 



d'q \ dq q 



J^ + -:r-^ = 0, (22) 



dr- r dr i- 



and the equation of continuity is satisfied identically. 



C 



The integral of (22) is q = — + C'r. 



If a is the radius of the inner, and b that of the outer cylinder, and if q,, q^ are the 

 velocities of points close to these cylinders respectively, we must have q = q, when r = a, and 

 q = qi when r = b, whence 



If ab I 

 9 = f^^_ ^, { (bqi - ag.) — + (bq, - aq,)r) (23) 



If the fluid is infinitely extended, 6 = co , and 



q a 



9i »• 



These cases of motion were considered by Newton, (Principia, Lib. ii. Prop. 51.) The 

 hypothesis which I have made agrees in this case with his, but he arrives at the result that 

 the velocity is constant, not, that it varies inversely as the distance. This arises from his having 

 taken, as the condition of there being no acceleration or retardation of the motion of an annulus, 

 that the force tending to turn it in one direction must be equal to that tending to turn it in 

 the opposite, whereas the true condition is that the moment of the force tending to turn it 

 one way must be equal to the moment of the force tending to turn it tlic other. Of course, 

 making ttiis alteration, it is easy to arrive at the above result by Newton's reasoning. The 

 error just mentioned vitiates the result of Prop. 52. It may be shown from the general equations 



