Mr Sharpe, On the Reflection of Sound at a Paraboloid. 133 



36. We will now shew that by particular linear combinations 

 of the two solutions V 1 and V 2 of equation (12) we are able to get 

 a single outgoing wave at points distant from the reflector, i.e. at 

 points for which v is large and A/v small. We will prove this 

 from the Definite Integral solutions of (12). From Art. 24 we get 



V x — — (e^ A + e~? nA ) I cos ( v cos x — A log cot -) dx .. .(87), 



and from (18) we get, putting v for u and — A for J., 



V 2 =- (e*" A + 6-4*^) x 



7T 



x I cos ( v cos x — A log cot - j . log (v sin 2 x) dx. . .(88). 



[It must, however, be carefully noticed that in this article we 

 are using A and v in the sense of the second part of Art. 6, that is, 

 A is for A/p and v is for pv.] 



Put 7j for the integral in Vj and I 2 for the integral in V 2 . 



From Art. 24 we get the value of I 1 for points for which A/v is 

 small. For brevity put 



v 1 for ( v cos a. — A log cot - j . 



Then for such points, by Art. 24, 



J 1 = {nrj^f (v cos a) ""* (cos v x + sin Vj), 



and 1 2=] cos f v cos x — A log cot - J . (log v + 2 log sin x) dx. 



It will be found that when v is large and A/v is small the most 

 important part of the integral I 2 is near x — a, where sin 2 a = A/v. 

 Accordingly following the method of Art. 24, that is, putting in I 2 , 

 x=a+<j> and integrating with regard to </> from —0, to +■ <£ 2 

 where fa is a small finite angle, and expanding log sin x or 

 log sin (a + <£) in powers of <f) as far as (fy 2 , we shall have 



I 2 — I cos (v x — v cos a . (j) 2 ) x 

 J -</,, 



x (log v + 2 log sin a + 2<f> cot a — (f> 2 cosec 2 a) d<p ; 

 putting v cos a.^ 2 = i|f 2 we get in the limit 



I 2 = I cos (v, - -f 2 ) . log A ^— - t 



J -oo (« cos a)- 



— cosec 2 a cos («i — v cos a . </> 2 ) . 2 cZ</>. 

 J -(f), 



