Mr Sharpe, On the Reflection of Sound at a Paraboloid. 137 



The two foci of reflection F l and F 2 are indicated in Fig. 10, 



1"84 

 where 0F X = 0F 2 and 0F 1 = and velocity at F x = 2p x -58 = 



velocity at F 2 , but in an opposite direction. 



We see therefore, that (for a given reflector) as p increases 

 (that is for higher and higher notes) the two foci F 1 and F 2 draw 

 nearer together and the velocity at each one increases. 



It is readily seen from equation (15), putting A = 0, that the 

 velocity at vanishes. Thus when p becomes large, the velocity 

 in the neighbourhood of tends to become discontinuous, but as 

 p has finite limits (Art. 5) actual discontinuity cannot be set up. 

 If we make OL' = OL and draw the parabola L"L ', we can show 

 from the formula given for the velocities in Art. 21 that the motion 

 in the space L"LL' is symmetrical with regard to the lines L"0 

 and LL'. (Of course only half the figure is drawn.) 



42. To complete the case of A = we have now to evaluate 

 the definite integrals in (33). This was done very imperfectly in 

 Art. 15. The following seems a better way. It is true that the 

 evaluation can be performed in a shorter way, but it will be found 

 convenient for future investigations, first to suppose A finite, and 

 then as a particular case to put it = 0. Let I l3 I 2 be the two 

 definite integral solutions of the equation 



d'X dX 



+ ir + {x + A) X = 



LvrV \Ajdb 



.(98), 



•It 



and let I 3 = j cos (x cos 6 + A log cot ^ j . log sin 2 #rf#, 



and let X lf X 2 be the solutions in series, the latter involving log&\ 

 Then by (18) 



Xj - - (eW + e"^-') I, and X 2 = - U*" A + e~^ A ) (I, log x + J 3 ) 



7T 7T 



(99)- 



