received Theory of Sound. 57 



inversely as the square of the distance ; but these conclusions 

 do not in the slightest degree militate against each other. 



If we suppose b to increase indefinitely, the condensation or 

 rarefaction in the wave which travels towards the centre will 

 be a small quantity, of the order 6~', compared with that in 

 the shell. In the'limiting case, in which b= oo , the conden- 

 sation or rarefaction in the backward travelHng wave vanishes. 

 If in the equations of paragraph 3 we write b + x for r, b(s{x) 

 for/'(r), and then suppose b to become infinite, we shall get 

 as=<T(a;), v = <t[x). Consequently a plane wave in which the 

 relation v=as is satisfied will be propagated in the positive 

 direction only, no matter vf\ie\herj'cr{x)dx taken from the 



beginning to the end of the wave be or be not equal to zero ; 

 and therefore any arbitrary portion of such a wave may be 

 conceived to be isolated, and being isolated, will continue to 

 travel in the positive direction only, without sending back any 

 wave which will be propagated in the negative direction. This 

 result follows at once from the equations which apply directly 

 to plane waves ; I mean, of course, the approximate equations 

 obtained by neglecting the squai'es of small quantities. It may 

 be observed, however, that it appears from what has been 

 proved, that it is a property of every plane wave which is the 

 limit of a spherical wave, to have its mean condensation equal 

 to zero; although there is no absurdity in conceiving a plane 

 wave in which that is not the case as already existing, and in- 

 quiring in what manner such a wave will be propagated. 



There is another way of putting the apparent contradiction 

 arrived at in the case of spherical waves, which Professor 

 Challis has mentioned to me, and has given me permission to 

 publish. Conceive an elastic spherical envelope to exist in 

 an infinite mass of air which is at rest, and conceive it to ex- 

 pand for a certain time, and then to come to rest again, pre- 

 serving its spherical form and the position of its centre during 

 expansion. We should apparently have a wave consisting of 

 condensation only, without rarefaction, travelling outwards, in 

 which case the conclusion would follow, that the quantity of 

 matter altered with the time. 



Now in this or any similar case we have a perfectly definite 

 problem, and our equations are competent to lead to the 

 complete solution, and so make known whether or not a wave 

 will be propagated outwards leaving the fluid about the enve- 

 lope at rest, and if such a wave be formed, whether it will 

 consist of condensation only, or of condensation accompanied 

 by rarefaction : that condensation will on the whole prevail is 

 evident beforehand, because a certain portion of space which 

 was occupied by the fluid is now occupied by the envelope. 



