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



with the numerical results of Helmholtz's theory and with ex- 

 periment. It must be understood that these numerical results 

 can be obtained in several different ways ; and other condi- 

 tions, such as knowledge of the coefficient of extinction, or 

 proportion'^of energy failing to reach the source after reflexion, 

 and knowledge of the actual form of motion, must be applied 

 to distinguish between the hypotheses. The calculation re- 

 duces itself, according to the methods of the last note, to the 

 evaluation of 



where S is the surface of the wave-front. 



To determine this surface, 

 let a be the radius A C of the 

 circular end (Sq) of a cylin- 

 drical tube, r the radius with 

 which the spheres are de- 

 scribed whose sections are the 

 circles shown, r being also the 

 distance through which a 

 given disturbance is supposed 

 to have travelled since it left 

 Sq. It is required to find the 

 surface of revolution about the axis of the tube traced out by 

 the outer limit of the circles shown. 



The curved half-ring surface traced out by the semicircle 

 whose centre is A is easily found by the theorem of Guldin ; 

 it is 



(2r \ 

 1- a ) = 47rr^ + ^ir^ar. 



Adding to this the plate ird? which lies parallel to the circular 

 end of the pipe, the whole surface is 



47rr^ + ^TT^ar + ird^. 



Or, if we suppose the flange (indicated by the dotted lines at 

 the sides) to be introduced, the curved part has half the value 

 above stated, and the surface for the case of hemispherical 

 divergence becomes 



^irr^ + TT^ar + 7ra^. 



As all our proceedings are approximate, I shall not introduce 

 the above expressions for the surface into the integrals, but 

 employ functions of a simpler type, conditioned to have the 

 value TTci'^ at Sq, a value determined from the above at r=:a, 

 and a value approximating to that of sphere or hemisphere for 

 large values of r. 



