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



We will now suppose x large and Ajx a small fraction. In I x 



x — t 

 put cos 6 = , and we get 



A , t ) dt 



Jl = /"^ {*"*"¥ l0g .2^=i} 



2a; -*J "#(2a?-«)*' 



ApplyiDg to this a method suggested by Art. 405 of Todhunter's 

 Laplace's Functions, we have 



A lo %-^h~t = A log * ~ A log 2x " ^ log ( x - i) • 



Here for A log ( 1 — — ) we may put zero, when t is not large, and 



when t is large, the corresponding elements of the integral are of 

 no account, because then t~% is small. 



For the same kind of reason, in the multiplier, for (2a; — i)% we 

 may put (2a;)i 



Then, when x and A have the kind of values supposed, 



In a similar way we get 



f x [ , A . * I . 2te - £ 2 <ft 



A = cos - a; — t — -z- log = -\ . log — . t=- — T , 



3 Jo I 2 6 2x-t) 6 x 2 {1tx-Vf 



and when x is large and Ajx small, this becomes 



J s = J o cos \x + j\og2x-t-j log t\ . (log 2t - log a?) . ^— ^ 



f™ ( A A ) dt 



.-. 7 2 =J o cos|a;+ ¥ log2a ! -£- ¥ logi|.log2i.-^ r 



Put £ = 2ip and we get 



r r { A, „ , „ , , , 1 2cfyr 



/j =1 cos ja; + — log a; — 2-v/r 2 — ^. log yfr> . — ^- , and 



I 2 = I cos \x -f -=- log a; — 2-fr 2 — J. log yjrl . (log 4 + 2 log ty) . — p 



7 /""cos 

 Now put 7^ = I • (2>/r 2 + J. log i/r) . dty, and we get 

 1 6 Jo sin 



a^ Jj = 2 cos ( x + — log a; j . I 5 + 2 sin (x + — log x\.I 6 . . .(100), 

 f a; + — log x).I 5 + sin (x + -^ log a; J . I, 



x i I 2 = 2 log 4 



cos 



+ 4 cos {as +j log a?) . -~ -4sin (^ + - log A ~ ...(101) 



