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



nearer the geometrical focus, the larger be p, that is, the higher the 

 note experimented on. 



18. We observe from Art. 10 that for a given value of I we 

 have an infinite series of values of p from small to large, for which 

 A = 0. It seems very likely then that for intermediate values of p, 

 whether small or large, we should always have some corresponding 

 small values of A satisfying (19), and that the solution for these 

 small values of A would resemble pretty much numerically that 

 for which A = 0, only involving more complicated definite 

 integrals. 



Before, however, leaving the subject of the zero and small values 

 of A we may notice another simple and interesting maximum pro- 

 position. From (15) supposing A not to be zero, and making 

 the requisite changes, we shall get 



2?) ^7) 2 



F/ = A + (2i? (A * " 1} + m> {A5 - 5A) + &c -' 



9 R v 



V 1 '" = ( £ y (As-5A) + &c. 



Suppose for a moment we can have A =1. What this entails 

 will be seen presently. But if A = 1, V" will vanish if v = 0, and 

 then V"' is negative. So that if A =1, V-l is a maximum at 

 v = 0, that is, the intensity of the sound is a maximum at the geo- 

 metrical focus of the reflector. From (19) we see that if A = l, 

 then (Art. 6), restoring the original notation, we shall get 



cos 6 sin ( pi cos + log cot ^0) d0 = 0. 



If we regard I as given, this equation will give us an infinite 

 number of values of p, but inasmuch as p, in the case of all 

 audible sounds, is restricted between definite limits -^ < p < 62, we 

 see that we shall only get a finite number of sounds to satisfy the 

 equation, but of course these sounds range from the lower to the 

 higher parts of the scale — limited of course by the value of I 

 chosen. For all these, A being = 1, the intensity of the sound is 

 a maximum at the geometrical focus. Of course, as in Art. 12, it 

 does not follow that the geometrical focus is the point of absolutely 

 greatest sound intensity. 



