Mr. Ei. H. M. Bosanquet on tlie Theory of Sound. 135 



That is, for small values of k^ r approximates to a value of 

 about "Sa under the circumstances of motion under investigation. 



In the present unsettled condition of the experimental 

 values for tubes which are very long in proportion to their 

 diameter, it would be rash to pronounce definitively ; but, so 

 far as I know, values of r so low as this are not well estab- 

 lished, although, as far as my own experiments go, I cannot 

 say that 'Sa is beyond the range of experimental error. If 

 for the present, however, we admit that the value '56a repre- 

 sents the present experimental results more nearly, we learn 

 that the divergence of the reflected stream of energy, so far as 

 it lies within the issuing stream of air, does not take place in 

 the manner above supposed, i. e. it does not spread freely 

 with hemispherical divergence, but behaves more, as far as 

 amount of divergence goes, so as to agree with the law of 

 reflexion from an image of the source. The analytical dis- 

 cussion of the actual motion is a matter which I do not pro- 

 pose to attempt ; but the following considerations will show 

 the general nature of the motion regarded as a modification 

 of spherical divergence, and will give some idea of the extreme 

 complexity of this part of the problem. 



In spherical or sectorial divergence we saw that the re- 

 flected energy formed a convergent stream, and suggested 

 that divergence might be prevented by the symmetry (i. e. all 

 the motions necessary for divergence may be reduced to equal 

 pairs in opposite directions). In the present case of diver- 

 gence through a cone of small vertical angle, the constraint 

 at the exterior surface of the cone is entirely removed, the 

 reflected energy close to the surface diverges freely in the 

 hemispherical form, and gives rise to the inward flow tov/ards 

 the surface of the cone which takes place through the sur- 

 rounding air. But in positions near the axis of the issuing 

 cone there is but little, if any, less constraint than in the case 

 where the sectorial form is maintained by enclosure within 

 walls ; so that this portion of the reflected stream does not 

 get away freely, as is supposed in the last investigation. 



The results up to the present point are shortly as follows: — 



Referring the case of sound issuing from a tube to the case 

 of continuous flow instead of the case of electricity, when the 

 wave-length becomes great with respect to all the dimensions, 

 we get a case which we can study experimentally. 



The experimental study teaches us that the form of velo- 

 city-potential derived from the hypothesis of a tendency to 

 uniform spreading does not represent the form of diver- 

 gence for outward flow, but that it does represent the form 

 of divergence for inward flow. 



