PROF. H. M. MACDONALD ON THE DIFFEACTION OF 



Writing 



= -- sn cos 



sn 



the appropriate approximations for R, E, l5 <, and (^ are 



R = sec a, </> = z cos a j*Mr+(n+i)*, where z sin a = 

 R 1 =seca 1 , <f> l = z l cosa l ^nir + (n + ^)a. ll where Zi sin a : = 



lience approximately 

 B! = -ij sin 9 cos L/? 2 (n + )* sec 1 -' 4 sec-' aie 'C--c.- ri co.. 1 +(,+vj(.-.,)] . J,{(2n+ 1) sin ^}. 



K 



Kemembering that the oscillations of Ji{(2n+l) sm|0} depend on e (2+D'^v.^ tne 

 principal part of S 1; arises from the terms in the neighbourhood of the term for which 





z cos ?! cos 



J) (a i) 



sin 



vanishes; for this a a, is small since 9 is small, and therefore, unless r is nearly 

 equal to ;], which means that the point is close to the oscillator, ii>-\-\ is small compared 

 with ~ for the terms that contribute the principal part of S,. Hence approximately 



z cos a 



: ! cos 



a = z + 



and the principal part of Sj is equal to the principal part of 



-^ sin 6 cos J 



K 



sn 



that is, the principal part of S : is given by 



S, = -L- sin cos 



/f'>'l 



sn 



therefore 



or 



n i . , 2sin4tf -.Hii^M 



b, = --- - sin 9 cos f Ve' ( - '>> -;_ 2 -rrr, e l*"-*!- 1 ) 



/c?r (z L 2j i ) 



q _ LKT 2 SJn 2 - (r,-r+ |2LdBV*) 



^1 ~" \^ C> . 



Appendix, Relations (ii). 



