120 PROF. H. M. MACDONALD OIS T THE DIFFRACTION OF 



4. When 9 is nearly equal to IT the appropriate approximation for P n (/u.) is 



given by 



P B (JU) = cosn7rJ {(2n+l)sml(7r-0)}, 



or, writing IT 9 = 6', 



P n (^ = cos mrJ {(2w+ 1) sin 0'}, 

 and 



= - (n+J) cos WTT- cos ^ cosec 6'J l {(2w+ 1) sin 1(9'} ; 



O LC 



hence 



= ---5 sn 



1 



when ?<?-!, where to the order required the terms for which n + ^ exceeds z by a 

 quantity of the same or of higher order than z' 3 can be neglected. Writing 



j = - 2 sin & cos l0V(n + ) 2 cos 



sn 



the terms of this series, for which w + |- is less than z, and zn\ is of higher order 

 than z 3 , are at most of the order (T)* / '* (Wi)~ 8 i/r 1} and therefore their sum will be 



negligible -in comparison with ^ unless -y [_<f> ()>i + niT (2n+ 1) sin |~0'] vanishes or is 



a multiple of 2n- for some value of n in the series. Using the appropriate approxima- 

 tions for <j) and ^ 1; this requires that a MJ + TT is small or nearly equal to a multiple 

 of 2vr since & is small, and this is impossible, for a and ctj are each less than \TT. The 

 terms of the series S l5 for which \z n 1-| is of lower order than z ! \ are at most of 

 the order (/c?-) 5 ' 3 (/f?-,)" 2 ^, and their sum will be negligible in comparison with ^ unless 



-j-[< $iti7r(2w + l) sin ^-0'] is very small or nearly equal to a multiple of 2ir for a 



value of n in this series ; substituting the appropriate approximations for <f> and <], 

 this requires that \TT a, 77- is small, which is impossible, for MJ is less than ^77. 

 Hence Si is negligible in comparison with t//j. The same result follows when ?>?*!. 

 Again, writing 



S 2 = - -i^ sin & cos 10'2 (n + i) 2 cos n ff R I /.R i V< 8 *.+ 2x .-*-*,> ^ {2n+ 1 sin 



the terms of this series, for which n + ^ is less than z , and z a n^ is of the same or 

 of higher order than z ' /3 , are at most of the order (/ca) 3/1 (KT^^ and therefore their 

 sum will be negligible in comparison with t// t unless 



sn 



vanishes, or is nearly equal to a multiple of 2ir for some value of n in this series. 



