Mr Sharpe, On the Reflection of Sound at a Paraboloid. 105 



I shall always call this A, "the original A" (see Art. 6). 

 Throughout a great part of the following investigation, i.e. from 

 Art. 19 to end, it will be supposed to have the same large value. 



It is easy to shew that the velocity in the direction Pp, fig. 2, 



or normal to the paraboloid u = a constant is -j- ( — j , and as the 



sound has to be reflected from the paraboloid u — I = 0, we must 

 have 



dF n i, 7 



^— = when u = l, 



du 



which leads to —— = 0, when u = l (10). 



du 



It will be seen that the condition (10) determines a value or 

 series of values for A. It remains to solve the equations (8) and 

 (9) with the condition (10), and to find what kind of sound 

 motions they represent. [It is important to notice that if we 

 take as our units 1 foot and 1 second of time p will vary from 

 about ^ for low sounds to about 62 for high sounds.] 



6. In nearly all that follows it will be found a great convenience 

 to use new independent variables, and to put 



pu = u, pv = v, pi = I' and A/p = A' (11)- 



(8) and (9) then become 



^ +d ^ +(v '-^ r=0 < 12 >- 



from which p has disappeared and (10), the condition of reflection, 

 becomes 



~ = 0, when u'=V (14). 



du 



In nearly all that follows we shall for convenience drop the 

 dashes, but we must be always careful to remember what the new 

 u, v, I and A mean, and in all final results we must remember to 

 restore to the symbols their proper meaning. It will be noticed 

 that when we are experimenting with the sound for which p=l, 

 the old and new symbols are identical. [The note indicated has 

 857 vibrations in a second, and is very nearly the F just above the 

 middle G of a piano.] 



