140 Mr Skarpe, On the Reflection of Sound at a Paraboloid. 



to the late Sir G. G. Stokes. It will presently be obtained in 

 another manner. It will be noticed that (106) is here obtained 

 without the use of complex quantities. 



From (103) and (104) it follows that when x is large and Ajx 

 small, 



A 



C 4 Xx — 2 X 2 



cos ( x + — - log x 



x* 



A 



sin [x + ■=■ 



C 1 X 2 — (jzX± = 



log x ) 



x% 



.(107). 



If smaller terms are required than those given in (103) they may 

 be got thus. In (98) put xi X = W. Then 



d 2 W ( . 1 , .,, 

 x -j-r + (x + A + — )W=0. 

 dx- V 4<x/ 



If in this we put 



-,Y . W-- [A + ^ + ^ 



+ &c. ) cos (#+-=- log 



+ { B ° + i + J + &c -) sin ( a + 2 log x ) ■ ■ - (108) ' 



we shall find that A 1 A 2 , &c, B X B 2 , &c, can all be obtained in 

 terms of A Q and B by the following equations : 



A 





2n5, 



(271-1)5,^=0, 



In (n - 1) - -^ +-i| A-i + 2n.4 n 4- - (2n - 1) ^ - 0. 



If in the above we put A , B = C\, G 2 respectively we get X 1 

 and if we put A , B Q — C 3 , C 4 we get X 2 . The above series are 

 semiconvergent if x is large and Ajx small. 



43. We will now show that I 5 and I 6 depend upon 



V2 + 2 



The following is, if I remember rightly, practically Stokes's in- 

 vestigation. 



Put 1= I" e ii2x2+A]6gx) x dx. 



Jo 



In it put x = yK Then 



1 f°° 2% + -^ log?/-* log 7/ 



I = 2j € xdy - 



