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



now du vv . ,, , . -p. 3<A 



v- u = 1, that is R_r = i 



oz Oz oz 



therefore 



g^v "+'/ 2 ( 12; )/ - 



and writing ^-~ = tan x , this becomes 



|-{2 1/2 K n+1 , (tz)} =-2- 1 V / ^- I/ '<" +1/ >"-* l -"R-v.8ecY. 

 82 A 



Similarly, 



{z' 8 J.,j.iy (z)} = 2' s 7r~'*R~ '' cos (<&+Y) sec v 



8s A 



and therefore 



t^" v n + '., \'*^^/] / ^ l^^ m ^n-t-''.>\*"*' v /l ^ COS 



oa 



Hence the relations (4) and (5) become 

 1 



a , 



1 ! i 



i 



sn - ie -' cos 



when ?'!>;, that is 



2iK'l\~ 1 OfJ, 



when rj>?', and 



^ = ^2(2n+l)R 1 v 'R 1/a {e'-* ) '+e (3 * +2 *-*-*'>'}(l- /1A 8 )^=. . . (7), 



j-iK i -^ 1 (J LJL 



when r>r!. 



2. The value of \|/ at any point will now be compared with the value of i/ij at the 

 same point, and, the radius a of the sphere being supposed to be great compared with 

 the wave-length 2vr//c of the oscillations, it will be sufficient to compare the principal 

 parts of \jj and i//i. Now 



that is 



hence the principal part ^ of T//J is given by 



In calculating the principal part of i// it is convenient to consider first the contri- 

 bution of the terms for which n + ^ is greater than the least of the two quantities 



