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

 In the above case, 



?7?^ = 4, n\ = 2. 

 Tlieu equating the result of the above formula to — 2? "^'^ ^^^ 



r = '85a, which is a little greater than the value (•82 a) given 

 bj Lord Rayleigh. 



The two values given by these two hypothetical forms of 

 motion lie on opposite sides of the true value for hemispherical 

 divergence ; we may conclude therefore that the true form is 

 intermediate in character between the two. 



We may apply the above formula to get a more exact re- 

 presentation of the hypothetical complete divergence without 

 flange. Here the surface when 7' = (2 was to be W'^uro?. We 

 will put 



m\ = l, nl = 4, 

 making S =ll7ra^: whence we find 

 r = '674.a, 



which represents closely the experimental value I adopt ('55 a). 



That it may be understood clearly how it is that I do not 

 consider the correspondence of the values of this theory with 

 experiment to prove the hypothesis of uniform spreading, I 

 proceed to point out that the conceivable forms of divergence 

 are infinite in number, and that for every one a law of diver- 

 gence of the reflected elements may be imagined which will 

 give the results in question. To test whether any one form is 

 the real one, it is necessary either to investigate the form by 

 direct experiment (which is the course I have selected), or to 

 determine experimentally the degree of resonance, or to pro- 

 du<ie a complete theory determining the point (which in my 

 opinion cannot at present be done). Such a theory has first to 

 account for the fact that air issuing under low pressure from 

 the end of a tube diverges in a cone whose vertical angle is 

 approximately 20°, and excites in the surrounding air streams 

 of the nature which have been described. Without dealing 

 generally with the infinitely various forms of divergence, I 

 w^ill touch upon certain cases which illustrate the passage from 

 the one form to the other ; and it will be convenient to asso- 

 ciate these with the simplest forms which can be supposed to be 

 assumed by the law regulating the divergence of the reflected 

 elements of energy per second. 



The divergence of the reflected elements could conceivably 

 take place in several ways : — First, according to the law which 



