Mr Sharpe, On the Reflection of Sound at a Paraboloid. 135 

 we get 



1 / \^A . 1 A\ C0S V l ■ IT 



7T V* 12 



9 A sin ^ 



— — -(- Filog^H- V s ) +w-*r i sin | 



.(89), 



- (e*^ + e"**- 4 ) — , sin — = 

 7r y y* 12 



9^4 cos J 

 4 



.(90), 



= 2r(1) (Vi ^g 4 - F 2 ) - 7T-4 F x cos -... 



and we see by Art. 5 that the motion of the single outgoing wave 

 is given by such a form as 



F=- (e^ A + e-* nA ) sin — r . v 1 , 4- — ' . 



7T 12 V* 



Remembering (Art. 24) that for v large and A/v small, ^ is 

 approximately given by 



v 1 = v-%Alogv + G 1} 



this becomes 



1 7T 



F=- (e^ A + e~^ A ) sin r— 



7T 1 J 



cos (y — ^A log y + Cj — 2pa£) 



.(91). 



37. We have now the solution of the whole problem, which 

 may be expressed thus 



F=Q cos 2pat + P sin 2pat, 



where 



9A 



Q=U 1 



P^U, 



-9Alo g A V3 ^Ay 

 J 2V2T(|) 2 J 1 ^2V2T(|) 



x K 



9^ log J. 



-Itt-HF,- 



9A 



x K 



(92), 



.(93). 



: _l2V2r(§) 2 " J ri 2 v / 2r(|) 



We see thus that the factors mentioned in Art. 33 are deter- 

 mined. 



We have already found the value of F for large values of v and 

 small values of A/v, but we might now go on to find its value for 

 any other values of v, say, for instance, small values, in which case 



