-132 i>ROF. M. Itf. MACDONALD ON THE DIFFRACTION OF 



where D is d/dt and t is put equal to zero after the operations are performed. This 

 is equivalent to 



e mD l 



S m = g - I-^n+) 



\s ~~ J- 



or 



^'" = t) gP_i ( e '" D ~ -0 u "+ 

 which can be expressed by 



fm T) 

 Q 11 rlt * 



"' ~ 75 r +* 



o ~ J. 



The present object is to replace this integral by expressions suitable for calculation 

 in different cases. The first case is that in which u n + f contains an exponential factor 

 e at , where 2&7rt (k an integer) does not vanish or is not very small. Writing 



where w K+t contains no other exponential factor of the form e at , 



D D a( D + 



_ rtt - ^ QI 'V/1 - P "111 



(J H^_1 " + ( ~~ gU_l "+ " g D + a_J " + 



that is 



or 



hence. 



that is 



D v J_ _d k I 1 \ d 



Therefore 



whence 



_ 

 o!aa*\l 



The sum S OT is thus expressed as the difference of two series which, in general, are 

 diverging series, but they can be used for approximate calculation if the last term 

 taken is within that part of the series which converges. When the series S m 



* Of. BOOLE, ' Finite Differences,' ch. v. 



