x(x x 



Mr Sharpe, On the Reflection of Sound at a Paraboloid. 139 



We have now to evaluate I 5 and I 6 . We shall show that they 

 can be expressed as Gamma functions of a complex, but before 

 doing this we will first establish a relation between them. It is 

 easy to show from (98) that 



x A = q no2^ 



dx 2 dx ) 



where G is some quantity independent of x. We can determine 

 its value from (15) and (16) by putting x, X 1} X 2 for u, U ly U 2 , 

 then putting for dX 2 /dx and dX-^/dx their values, and making x 

 vanish. We then get (7 = 1. Of course (102) may be regarded 

 as a First Integral of (98). If x is large and Ajx small, we can 

 show (Art. 23) that the leading terms in X x and X 2 have the 

 following forms 



a;*X, = Cx cos ( x + -=■ log x) + G 2 sin ( x + — log x 1 



v^X 2 = C 3 cos ( x + ^- log x j + C 4 sin ( x + ~- log x J 



,..(103), 



where G 1} G,, G 3 , C 4 are certain functions of A. From these last 

 equations finding the values of dX^jdx and dX 2 /dx (retaining 

 only the largest terms) and putting them in (102), it will be found 

 that we shall get 



0,^-^ = 1 (104). 



Comparing together the equations (99), (100), (101), and (103) 

 we shall get 



C x = - U*" A + 6-W) x I 5 and G t = - U*" A + e~^ A ) x /„ 



IT 7T 



G 3 =i ( e W f e-^) lll 6 log 4 + 4 — 6 ^ 



7T V V " dA) 



and C 4 = - (e^- 1 + e~^ A ) ( 2I 6 log 4 - 4 ^ 5 



7T \ (X-fl. 



...(105). 



Put these values of C l3 G 2 , C 3 , C 4 in (103) and we shall get 

 8 



/ dl 5 d%\ 



\ l5 dA + U dA) 



7r-\ 5 dA ' 6 dA) (e- 4 + l) 2 ' 

 Integrating we get 



(i n r-+(/ 6 )^^x e J ri (io6). 



It can readily be shown that no constant need be added. 



It is important to notice that in equations (100), (101), (104), 

 (106) A may have any value. For the formula (106) I am indebted 



