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



The 1st integral on the right 



/7r\* . . log J. 



= (. o] (cos v 1 + sin Vi) - — - — -i . 

 \*f (v cos a) 2 



In the 2nd integral put v cos a . <f> 2 = i/r 2 , then the second 

 integral on the right 



= — cosec 2 



a I cos (Vj — ijr 2 ) . — — — 

 J -oo (v cos a) 



(v cos a)* 

 In this integral put i/r 3 = x when it 



/•OO 



= — % cosec 2 a (v cos a)~* x I cos (^ — #*) dec, 



J —00 



from Art. 28, 



Finally, 



(wCOSa) s / w\ __,,_. 



9 • / x cos U - - x §T (§). 

 3 sin 2 a V 3/ 6 6 



I* = « ( cos w i + sm v i) 7 a 



\2 / v ' (v cos a) 5 



2 (v cos a) - ^ 

 9 sin 2 a 



COS ( t>! — jr 



r(f). 



The last term 



2 v *(cosa)~* 



— ¥■ ^ X 



cos v 1 — 



r(|). 



Therefore the terms in 7 2 are both of the order of v~%. 



If we solve these simple equations so as to get vicosvj and 

 v~* sin v x in terms of I x and I 2) as a is small and becomes smaller 

 the larger be v (A being supposed fixed), we may, as we are in- 

 vestigating the state of things far from the focus of the reflector, 

 suppose a = in the limit, and we shall get 



x(7 1 log^l-/ 2 ), 



V 2C °T 1 -3r2T(!) 



V~% COS [Vi — -T-) = TT~* X I 1 . 



If we multiply each of these equations by 



( e £"-.4-|_ €~^ wA ), 



