196 DR. T. J. I'A. BROMWICH ON THE 



The leading parts of M, N are accordingly given by the approximations 



(6-6) 

 where 



KT 



.Q, N=-Bii 



KT 



-Q, 



O = 



and o- has the value given in (6 '51). 



The value of Q is approximately equal to the integral 



(6-61) 

 where 



Thus, approximately 



J_ v /(2ra 7r sin 0)' 

 HQ+X, a- = <r u +x 2 /(z cos 0). 



= <f" /(TO, cos &0\ 



' ,/(2n Tr sin 0) "V \ - 4 / 



2 sin |-0 2 sin ^0 

 Accordingly, to the same degree of approximation, we can take 



fc-cn\ (^2 1_ 2t; COB if 



(b DZ) ^ = -nZC, 



dQ 

 Then the components of force are given, as in (4'9), by 



(6'63) 



rj 



Z = -c/3 = -T 



3M 1 3N 

 30 sin 30 



3M . 3N 



= COS 



KT dB 



sin 



^ 

 30 



/cr 



where differential coefficients with respect to are small compared with those with 

 respect to 0, and so have been rejected. Thus, using (6'62), we have the approxi- 

 mations to the forces in the scattered waves 



(67) 



Y = +cy = cos e 2 "" 1008 * 9 -""", 

 2r 



Z = -c/3 = - sin 



e*" 0001 *-"", 



assuming that is neither near to nor to TT. 



When is small, the approximation to P n (cos 0) must be taken as 



and so 



P.(cos0) = J 

 sin 0P' B (cos 0) = (n + f ) cos f Jj { (2n + 1 ) sin J0} . 



