Mr Sharpe, On the Reflection of Sound at a Paraboloid. 129 

 Next at P 2 (fig. 4) by (56) and (61) the magnification 



As (Art. 6) A here stands for A/p and v for pv, we see that 

 generally at such a point as P 2 (fig. 4) the magnification is greater 

 for low notes than for high ones. 



Case of A large. Source near. 

 Sound-sending Reflector. 



33. In the following Articles LO (figs. 2 and 4) is supposed 

 to be a source of sound, or more strictly a line of sources, and the 

 problem now before us is to find the single wave of sound, which 

 we feel sure must at a great distance be the result of such a 

 supposition. We shall find that such a single wave is the result 

 of combining both solutions of equation (12). 



We shall in fact find (Art. 37) that F, the velocity potential, 

 takes the form {CiU 1 V 1 + C<JJiV^) x time factor where G x and 2 

 are certain special constants whose values are determined in Art. 

 36. At present however we shall for simplicity suppose F to be 

 of the form U 1 V 2 x time factor, V 2 being the second or logarithmic 

 solution of (12), and we shall see afterwards in Art. 37 what the 

 results of Art. 35 have to be multiplied by, to adapt them to our 

 present purpose. 



34. And first to find the strength of the source, and the law 

 of this strength at various points of LO. Using at present u, v 

 and A in their original sense (Art. 6) it will be found (Art. 5) 

 that the rate of flow of ah' normal to v = constant across a small 

 piece of the paraboloid made by the revolution round Ox of Pp 

 (see fig. 2) is equal to ^irdu x vdF/dv where F is the velocity 

 potential. But F=U 1 V 2 x (time factor). We will for brevity 

 omit the time factor. 



Therefore the rate of flow = ^irU^it x vdV 2 /dv. 

 But if we take this rate of flow normal to LO (that is v = 0) we 

 have by Art. 5 vdV 2 /dv= 1 if we use the 2nd or logarithmic solu- 

 tion of (12). [Of course when v = that is at all points of LO 

 vdVJdv^O.] 



Therefore the rate of flow normal to LO = ^LirTJ^du. 



JJ X will be the law of the strength, and the whole strength will 



= 27r( JJ x du. 



Jo 



35. We will now examine the value of CTj for various values 

 of u, and in connexion with this it is a matter of much interest 



