through Apertures in Plane Screens. 261 



The second is satisfied if M> = — M* ; so that 



+ m =^V m e -^ r dS, +r=-ffi™ ~ 



dS, . (9) 



making the values of ^r m and ^ p equal and opposite at all 

 corresponding points, viz. points which are images of one 

 another in the plane # = 0. In order further to satisfy the 

 first condition it suffices that over the area of aperture 



+«=-l, +>=!> • • • • (10) 



and the remainder of the problem consists in so determining 

 M/" OT that this shall be the case. 



In this part of the problem we limit ourselves to the sup- 

 position that all the dimensions of the aperture are small in 

 comparison with X. For points at a distance from the aper- 

 ture e~ ih '/r may then be removed from under the sign of 

 integration, so that (9) becomes 



t =^jj^ s > t^-^jjV^s- • (ii) 



The significance of (T^dS is readily understood from an 

 electrical interpretation. For in its application to a point, 

 itself situated upon the area of aperture, e~ ikr in (9) may be 

 identified with unity, so that yfr m is the potential of a distri- 

 bution of density y i r m on 8. But by (10) this potential must 

 have the constant value — 1 ; so that — iJ^^S, or Jj^ptZS, 

 represents the electrical capacity of a conducting disk having 

 the size and shape of the aperture, and situated at a distance 

 from all other electrified bodies. If we denote this by M, 

 the solution applicable to points at a distance from the aper- 

 ture may be written 



sj—ikr p—ikr 



<r =-M- , <r = M- . . . (12) 



* m r p r 



To these are to be added the values of % in (6). The realized 

 solutions are accordingly 



cj> m = 2 cos nt cos hx - M C ° S ^ - h ^ , . . (13) 



</> p = M C0S {nt - kr) (1*) 



The value of M may be expressed * for an ellipse of semi- 



* ' Theory of Sound/ §§ 292, 306, where is given a discussion of the 

 effect of ellipticity when area is given. 



