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



We thus get two independent real solutions of (12). Con- 

 fining ourselves then to the leading term we may write approxi- 

 mately 



B G 



V 1 = -r cos (v — \A log v) + -i sin (v — \A log v) (56), 



where B and G are constants which must be determined by com- 

 paring (56) with the definite integral solution of (12) adapted to 

 the case of v large. Before doing so, however, we may notice that 

 the series (55) is adapted also to the case of A large as well as v 

 large, provided that A 2 /v be small. This remark is important 

 because, whilst using large values of v we may at the same time 

 use large values of A if the above condition is satisfied. 



24. We now proceed to find the value of B and G in (56). 

 From (18) we have 



1 /"i"" / x\ 

 V x = - (e%" A + e~** A ) J cos f v cos as — A log cot - ) doc (57). 



We must find what this becomes for large values of v and 

 moderate values of A and compare the result with (56). 



If my memory serves me right, I am indebted to Sir G. Stokes 

 for the following idea. At any rate he employs the very same 

 method in Note * on Art. 9 of his "Numerical Calculation of a 

 Class of Definite Integrals and Infinite Series." 



Trace the curve in x and y, whose equation is 



y = v cos x — A log cot - , 

 whence dy/dx = — v sin as + 



sins 



from x = to x = tt/2. It will have some such form as is found in 

 fig. 3. dy/dx will be everywhere numerically very large except in 

 the immediate neighbourhood of Q, where there is a maximum 

 ordinate. If a be the value of x at Q 



sin 2 a = A/v (58). 



On account of the rapid fluctuation in other parts, the most 

 important part of the integral for v large will be for a small range 

 of values of x to the left and right of Q. Accordingly to get the 

 value of V-l for v large, put in (57) x = u + <f), and integrate with 

 regard to </> from — fa to + fa where fa is a small finite angle. 



If/ be the integral in (57) we shall get, retaining powers of <f> 

 not beyond the 2nd, 



f* 1 T a 1 7 



/ = I cos v cos a — A log cot ~ — v cos a . fa dcf). 



