Mr. G. G. Stokes on a difficulty in the Theory of Sound. 355 



densed air comes to occupy, in the rarefied state, a length 

 (»+w% so that we must have p« =p'(a -f-w/). Next, if we 

 take two sections s v s 2 , the first to the left and the second to 

 the right of s, and suppose the first to remain at rest, with the 

 fluid in which it is situated, while the second moves, along 

 with the fluid in which it is situated, with the velocity tt/, since 

 the pressure on s x exceeds that on s 2 by « 2 (p— p'), the surface 

 of 5, or s 2 being for simplicity's sake supposed equal to unity, 

 there must in the time / be generated a momentum a?(p—p')t 

 in the fluid lying between s x and s 2 . But this will be the case 

 in consequence of the velocity w/ being communicated to a 

 volume at of air which was previously at rest and of density p, 

 while the state of rest or motion of the remainder of the air 

 between s x and s 2 has been unaltered, provided a 2 (p—p')=i^6p. 

 These two relations being satisfied, it appears that the motion 

 is dynamically possible. The two equations might have been 

 obtained at once from (2.) and (3.) by writing — » for » and 

 putting w = 0, but I have preferred deducing them afresh from 

 first principles in consequence of the novelty of the subject, 

 and the reluctance with which the conclusions that I have 

 arrived at are likely to be received by mathematicians. 



These conclusions certainly seem sufficiently startling ; yet 

 a result still more extraordinary remains behind. By solving 

 the two equations of the preceding paragraph with respect to 

 y and «, we get 



= \f 1 **- 



Now let p' vanish; thenw/ becomes infinite and evanishes. Hence 

 the rate at which the condensed air (which remains packed 

 like the combustible matter in a rocket) is discharged decreases 

 indefinitely as the space into which the discharge takes place 

 approaches indefinitely to a vacuum. Of course the velocity 

 of discharge becomes infinite, without which the requisite mo- 

 mentum could not be furnished. The quantity of air which 

 passes in a unit of time across a plane, of area unity, taken at 

 the positive side of the tube, is vJp\ which is easily seen to be 

 a maximum, for a given value of p, when p' — ^p. 



A similar paradox is fully considered by MM. Barre de 



* It is worthy of remark, that when p' is very nearly equal to p, and 

 consequently to' very small, the velocity of propagation » is very nearly 

 equal to a, to which it approaches indefinitely when w' is indefinitely dimi- 

 nished. Thus even this discontinuous motion offers no exception to the 

 theorem, at once proved by neglecting the squares of small quantities, that 

 in very small motions any disturbance is propagated in the fluid with the 

 velocity a. 



