PROF. H. M. MACDONALD ON THE DIFFRACTION OF 

 when 2-wJ is of an order higher than z ! \ and this is equivalent to 



u = R 1/2 sin <f>, v = R v - cos <f>, 



where 



R = sec a and <j> = zcosa nir+(n + $)a. ..... (ii.). 



When zn^ is of the same or of lower order than z' /3 , the part of the first integral 

 in the expression for v iu that is most important is that contributed by small 

 values of 6 ; the second integral now becomes of the' same order, and the most 

 important part of it is contributed by values of ^ near to zero ; hence 



-i fV' --"-'"> "-''^ (16+ (V ( ---''-''^-''<--* 3 c 



the remaining parts being of lower order, and therefore writing 

 = 6'W^C, V = G v '2~'", G'V 



it follows that 



L- o Jo 



where and , are large quantities, the first being proportional to 2 1/3 # and the second 

 to 2' :1 i// , and as only the most important part of VM is required, and the parts 

 contributed by large values of are negligible in comparison, and & may be 

 replaced by <x> . Hence all but the parts of highest order being neglected 



. . . (iii).* 



The values of R and <f) in this case have now to be obtained, and it is convenient to 

 take first the case where jn-f- + z| is small compared with z' /3 . The expression in 

 (iii) can for this purpose be expanded in ascending powers of /A which gives 



VM = "Z '-fa /3 7r i 2' 6 2, 



*=ori(Ar) 



that is 



For the purposes required it is sufficient to know the values of R and < to the 

 second power of /A, hence neglecting /x :i and higher powers of /x,, 



v = 2' / 3- ! Sr- 1 '''z 1/ ' [II (- f ) + 3/tII (-|)] cos 2 TT, 



u = 2 s/6 3~ 2/3 7r~ 1 V /6 FIT ( fl Sail ( i)l sin ATT cos i?r : 



* It can ])e verified that this integral is a solution of the differential equation -^ - 9/*y = 0, which is 

 approximately BESSEL'S equation for r~ 1/3 K n+1/ when | n + J -z \ is of the same or lower order than z /a . 



