Mr. K. H. M. Bosanqiiet on the Theory of Sound. 131 



would subsist if we treated each reflected element as an inde- 

 pendent source, and applied the Helmholtz-Rayleigh theorem 

 of reciprocity* This simply amounts to saying that it does not 

 matter which way a disturbance travels from one point to an- 

 other ; the fraction of the energy per second of unit surface 

 at the source v^^hich reaches the unit receiving surface is the 

 same, whichever of the two be source or receiver. In this case 

 it is easy to see that the above fraction, from unit surface to 



(gOv 2 

 ^ j , if we suppose the divergent stream to 



behave as in spherical divergence. (For the fraction of energy 



from Sq that reaches S is ^ ; and of this only the fraction -^- 



lies on the surface Sq equal to that of the source.) The frac- 

 tion of any energy-element from S that returns to Sq is then 



measured by -^. (For it is the fraction ( -^ ) taken as many 



times as the surface Sq goes into the surface S.) The energy 

 reflected from surface S to Sq is consequently proportional to 



^, the reflected element itself being proportional to -^). 



This is on the hypothesis that the reflected element behaves 

 as an independent source. 



Secondly, it may be imagined that the energy-element di- 

 verges from the reflecting surface S in the same manner as if 

 it were diverging from an image of the source Sq placed behind 

 S. The motion would be, from this point of view, regarded 

 as a directed one, of the nature of projection of a number of 

 small particles from Sq which make their way by impacts. 

 For small distances from Sq, the surface S, in \^hich elements 

 of energy emerging simultaneously in paths nearly parallel lie 

 at any time, is approximately plane. The magnitude of the 

 surface % in which they would lie after a formal reflexion at 

 S, according to the ordinary law (angle of incidence = angle 

 of reflexion), = the plane section of the cone of divergence lying 

 at the same distance from S as S from Sq. Consequently 



5^ s ^^, ^^S^^ 



8 So Sq 



This surface S, which should be occupied by the energy re- 

 flected from S, lies in the plane of Sq ; and the fraction of 

 reflected energy that would reenter S is 



S ^ Sq _ SJ 



^ ^7^' ^s 



The reentering energy-element is here measured by -05 • 



K2 



