122 PEOF. H. M. MACDONALD ON THE DIFFEACTION OF 



and r t are so great compared with a that sin a and sin j can be replaced by a and 

 the principal part of S 2 is equal to the principal part of 



_L sin 9 cos iff 2 (w + i-)' e-'t- + --' +1/ '<" +1/ ^- 1+ " -H jj {(2n+ 1) sin 



"''I -oW* 



which is equal to 



cos 



When 0' is very small the value of the above integral is approximately 



IK?" 



which is equal to the value of i//j at the point. These results correspond exactly with 

 those obtained by the usual treatment of the effect of an opaque circular screen 

 interposed between a source of light and the point of observation, when the source 

 and the point of observation are both at distances from the screen great compared 

 with the radius of the screen. It has been assumed in the above approximation 



a 2 a 2 



that 1 is a small quantity whose square is negligible ; when this condition is 



A? ' \TI 



not satisfied the value of the integral involved depends on the quantity 



X /(K) (a + a t ) (tan a + tan i)~ V;! , 



where sin = n/r, sin j = /?'i, and diminishes rapidly as this quantity increases. 



5. When 9 is not small or nearly equal to IT, P n (n) can be replaced by its 

 approximate value 



2'Sr- 1 '' {(n + l) sin 0}- 1 /' cos {( 

 whence 



^ = (2n+ I)'/-' (77 sin 3 0)- 1 /' sin 



CtLL 



and therefore, when r <rj, 



sn 



approximately, that is 



which may be written 



/f = Si 82+8384, 



