124 PROF. H. M. MACDONALD ON THE DIFFRACTION OF 



If (XP = D, then 



and therefore 



^ cos ! r.cos a = D, r sin 6 = D sin aj, 

 S 21 = uc sin 2 0rD- V" D , 



that is for the points defined above S 21 = i/i. 



The same formulae (ii.) of the Appendix are applicable to the terms of the series S 3 

 and S 4) for which n + % is less than z and z ( + |-) is of the same or higher order 

 than z '\ The corresponding parts of the series S 3 and S 4 are given by 



lOQj 1 ^iTT t Qi.Il ' 



g _ (2iT)~ l '-s\n''t 



where z sin = z sin a. z v sin a t = n + ^. Now 



d 



{2~o cos a z cos a Zj cos 



|-) (2a a c^ + 0)} = 2a a 



and > for r>, a >a! for ?'!>, hence 2 a oti+^ is always finite, and 

 therefore, as above, the sum of the terms S 31 is of lower order than x//j. Also 



-j- {2z a cos a u z cos a z l cos i + ( + |-) (2a a a. l 0}} = 2a .a.u l $, 



and 2oo 



principal 



6 will vanish for a value of n between 1 and z Az ' /3 , provided the 

 point P (r, 6) satisfies conditions to be determined. For 

 let n u + -| = Kp, where p <a, and let (XQ be a straight line 

 through O 1 at a distance p from O, meeting the sphere in Q 

 (fig. 2), then the angle TQOj is a , where n + ^ = z sin a , 

 and the angle OOjQ is a ls where n + ^ = z l sin a^ Hence, 

 drawing QP in the plane OjOQ making the angle TQP 

 equal to , and taking any point P on QP, the angle QPO 

 is equal to a, where n + ^ = z sin a, and the angle POOj, 

 which is 6, is equal to 2a a a^ For any position of P 

 on the line QP the terms of the series S 41 in the neigh- 

 bourhood of the terms for which n = n contribute a sum 

 which is of higher order than any of them, and the 



S 41 is given by 



S 41 = (277-)~ 1/a sin' /2 6 n a + i^ sec' /2 a sec 1/2 



g'/2 'f 2 \ - cos ^ ~ , cos 

 J 



