132 Mr. R. H. M. Bosanquet on the Theory of Sound. 



Thirdly, it is tolerably clear that the conception of the last 

 case is correct so far as the projection of the air goes, but not 

 in regarding the reflexion as like that of particles. What 

 really happens is, that the disturbance in its advance leaves a 

 certain defect of pressure behind it as it expands ; and this 

 behaves likes a suction of air into a circular opening, the sur- 

 rounding air pressing into it pretty uniformly from all direc- 

 tions. We shall return to this case. 



First, then, according to the law of reciprocity, which treats 

 the reflected impulse like an independent source, the element 



of energy which reaches Sq from S is measured by ^ ; con- 

 sequently the " centre of phase " of the reflected vibration is 

 determined by the equation 





So 



= mi 



The following are the results for a few assumed forms under 



this law ; — 



r 



a 



for law expressed by 



7r(a^ + r2) 7r{a? + r'') | 



7r(a2 + r2) ^irr'' -726 



7r(a2 + r2) 4:7rP -^-rra^ -69 



7r(a2 + 27-2) 7r(aH2r2) '555 



7r(a2 + 2r2) 4.irr''-ira'' '53 



7r(a2 + 3r2) 47rr2 '45 



We see that this law, which we may call the reciprocity 

 law for the moment, requires a form of motion somewhat dif- 

 ferent from any we have hitherto come across. The motion 

 represented by the first entry, which gives the number of the 

 Helmholtz theory, certainly does not exist ; nor do really any 

 of the others. Yet the results are useful ; for these are func- 

 tions of the type of those which must express the motion. We 

 can draw at least one conclusion about such functions in 

 general, for which we have been already prepared. Compare 

 the cases in which the same form at r-=a diverges into hemi- 

 sphere and sphere. One such pair of values is '726, '69 ; an- 



