20 PEOFESSOE HOEACE LAMB ON THE PEOPAGATION OF 



in accordance with the requirements of continuity. Thus, with the allocation shown 

 in the figure, we shall have, for small values of 17, 



a' = - v /(2/ii ? )e- 1 ' > , & = i v/(& 3 - /i 2 ) "I 



-.... (86), 



H 

 approximately. 



Taking the integral (66) round the several contours, in the directions shown by 

 the arrows, we find 



r ( 2 3 - k* - 2 

 \ ~F77 



cos KX 



r x [ 0/2 _ 7.2 _ 0,"/3" 9/2 _ 7,2 I oV/C// 1 



i (,-**[ J /4 -- /t - Z,ct ft /(,_- Ic + Za. ft 1 r , / 



Jo l(2* - yt 2 ) 3 - 4^""^ (2^ - P) 2 + 4V'" J 4e C ^ 



J_ --*. f J 2 ^^ 3 _Z-_ 2a '/ S/ 2^-P + 2a^ 1 . ... , 



Jo [(2? - A; 2 ) 2 - 4CV y S / "" (2^ 3 - 2 ) 2 + 4^V/i / J 4<? ^ ' 



cos KX 



_________ 



o (2^ - /r) 1 + 16^ (C 2 - 



where, in the first integral, = k + iy, and, in the second, = h + irj. 

 The integral (69), taken round the same contours, gives 



tfa. 



= 2,7 sn 



, ,. c rr _^<_ -p^ i 



Jo l(2>-F) 8 -4Ca'/8' (2^-l 2 ) 2 +4CV^J 6 ^ 



+ Ste-*" f- -^^-C? ~ /i2 ) ^ 

 Jo 2 2 -F* 16 2 - 



* f __ V(2^J?*r?ds 

 Jo(2^-F)*+16*--A*F 



on the same understandino-. 



o 



The definite integrals in these results can all be expanded in asymptotic forms by 

 means of the formula 



and when kx, and therefore also kx, is sufficiently large, the first terms in the 

 expansions will give an adequate approximation. 



