Mr Sharpe, On the Reflection of Sound at a Paraboloid. 131 



this value. The question is not easy, so the following is put 

 forward with some hesitation. First trace the curve whose equa- 

 tion is (82) regarding U^ as the ordinate and u the abscissa. The 

 general form is represented in fig. 7, being the origin and the 

 u's being measured to the left of 0. 



01=1, and since from (15) dU 1 /du = — A when w = 0, that 

 accounts for the curve being drawn downwards from /. 



If the curve in the figure correctly represents U-^ especially 

 for small values of u, then the greatest numerical ordinate will 

 obviously be at 0. Next we put (as in Art. 19) z for Au and we 

 consider z as small as possible, and yet so large that we may safely 

 neglect its inverse powers {so as to make (82) a true approximation 

 to U). We have then approximately 



dU, sm(2z h -jir) d 2 U 1= cos(2^-^tt) 



U x will be a maximum or minimum if 



2(Au)l-l7r = mr (83), 



n being an integer. [(83) is of exactly the same form as equation 

 (40). Perhaps to distinguish it from (40) it will be well to write 

 (40) thus 



2(Al)i-lir=n a7 r] (84). 



The point G is obviously determined by putting in (83) n = 1. 

 But another very important question now remains, which is — 



If we put n = 1, is the value of u so obtained from (83) large 

 enough to make (82) a reasonable approximation to the value of 

 Ui? To answer this question we put x temporarily for 2 (Au)?, and 

 we remember that by Todhunter's Laplace's Functions, p. 318, 

 (82) is only short for the following 



^ = (^)' C0S ^-^ ) -{ 1 -^f (^J 



l 2 . 3 2 . 5 2 . 7 2 / 1 V . 

 + 4.8.12.16feJ- &C ' 



+ (£f sin {x - ^ ■ fir© " ^ QxJ + &c j • 



If the 1st term of this expansion can be used alone as a good 



l 2 / 1 \ 

 approximation to the value of U 1 we must have t~ ( o - ) sma ^ 



compared with 1. This condition is fairly satisfied by the value 

 of x (viz. 5-7r/4) which we get by putting n = 1 in (83). The final 



