through Apertures in Plane Screens. 267 



tion of the extreme forms, which includes all that is required 

 for our present purpose, starting from the conception of a 

 linear source as composed of distributed point -sources. If p 

 he the distance of any element doc of the linear source from 

 0, the point at which the potential is to be estimated, and r 

 be the smallest value of p, so that p <2 =r <2 + x 2 , we may take as 

 the potential, constant factors being omitted, 



, _C" e~ ik ? dx _ f e~ ik § dp 



^" Jo P ' )r S{P*-**) 



(36) 



We have now to trace the form of (36) when kr is very 

 great, and also when kr is very small. For the former case 

 we replace p by r+y, thus obtaining 



r Jo vy- v(^+y)' • ■ ■ [6n 



When kr is very great, the approximate value of the integral 

 m (37) may be obtained by neglecting the variation of 

 V(2r+y), since on account of the rapid fluctuation of sign 

 caused by the factor e~^ we need attend only to small values 

 of y. Now, as is known, 



P°° Q^xdx C" ^Mxdx _ //tt\ 

 Jo ~~^x~~ -J ~jx~~-\/\2), 

 so that in the limit 



in agreement with (35). 



We have next to deduce the limiting form of (36) when 

 form VCrj SmalL F ° r thiS pur P ose we ma F write {t in the 



The first integral in (39) is well known. We have 

 re-®? dp 



, i 7 Wr 2 kW 



8 2 2 ^2.3.4 2 ' * * 



,; [it 7 , k d r 2 "I 



In the second integral of (39) the function to be integrated 

 vanishes when p is great compared to r, and when p is not 



