114 Mr Sharpe, On the Reflection of Sound at a Paraboloid. 



and the condition of reflection (10) gives us 



2{Alf-\ir = nir (40), 



where n is a large positive or negative integer. 



We see thus that the large values of A are nearly proportional 

 to the squares of large natural numbers. We must also remember 

 (Art. 6) that Al in (40) stands for A'l', and that this again is equal 

 to the original Al. We see thus that the large values of the 

 original A are nearly independent of p, that is, of the note ex- 

 perimented on. We say 'nearly' for a reason which will presently 

 appear. 



We note also that for a given value of n, A will be the larger, 

 the smaller be I. 



We note also that the series (38) virtually ascends by powers 

 of u 2 /(Au)z that is, by powers of u^/A^. 



In order then that (39) should be an accurate approximation it 

 is necessary that ifi/A* should be a small fraction. Of course from 

 (40) the same must be true of $/A*. 



Remembering (Art. 6) what u and A here mean will accouut 

 for the use of the word 'nearly' above. 



The result (39) can also be obtained in a totally different 

 manner. See end of Art. 22. 



Before leaving that part of the subject connected more especially 

 with the values of U and d U/du we may notice that by Art. 5 the 

 velocity of an air particle normal to a paraboloid u = a constant is 

 [now using u and v in their original sense (Art. 6)] 



= 2V -r~ ( ) x (time factor) : 



du \u + vj v y 



when u = this velocity vanishes, shewing that there is no discon- 

 tinuity at the axis. Again, by Art. 5 the velocity normal to a 

 paraboloid v = a constant is 



.dVf 



dV t ii \ 2 

 = 2 Uj- i~-) x (time factor) (40a). 



Supposing that we keep on this particular paraboloid and approach 

 the axis, it is interesting to see whether this last velocity increases 

 or diminishes. Supposing u so small that in (15) the smallness of 

 the u overpowers the largeness of the A, then it is plain from the 2nd 

 term of (15) that as we approach the axis (u = 0) U x increases. 

 Further, as u diminishes (u + v)~^ increases, therefore on the whole 

 as we approach the axis the velocity (40 a) always increases, that 

 is, there is always a concentration of sound in the neighbourhood 

 of the axis. 



