through Apertures in Plane Screens. 269 



The condition by which *S? m is determined is that for all 

 points upon the aperture 



[**V m T>(kr)dy=-l, .... (44) 



where, since Jcr is small throughout, the second limiting form 

 given in (35) may be introduced. 



From the known solution for the flow of incompressible fluid 

 through a slit in an infinite plane we may infer that ^ m will 

 be of the form A(b 2 —y 2 )~*, where A is some constant. Thus 

 (44) becomes 



A[ (7+ .o g ^ + j;;i^] = -i. . ( 45) 



In this equation the first integral is obviously independent of 

 the position of the point chosen, and if the form of ty m has 

 been rightly taken the second integral must also be independent 

 of it. If its coordinate be tj, lying between + b, 



r+b \ ogrc j (/ _ p log (>n—y)dy P Mog (y-r})dy 



and must be independent of r). This can be verified without 

 much difficulty by assuming rj = b sin a,y = b sin 6 ; but merely 

 to determine A in (45) it suffices to consider the particular 

 case of 77 = 0. Here 



r+b \ g r dy _ p log y dy 



Thus 

 and 



= ^ (""log ( h sin 0) dO = ir log ($*)'. 



A ( 7 + log iikb)7i = -1, 



so that (43) becomes 



+ m= v+io g '(im)(M?f- ■ ■ (46) 



From this \fr p is derived by simply prefixing a negative sign. 

 The realized solution is obtained from (46) by omitting the 

 imaginary part after introduction of the suppressed factor 

 e lnt . If the imaginary part of log (\ikb) be neglected, the 

 result is 



. _ / 7T \* cos (nt — h — \ir) 

 Tm ~\Wr) y + log (ikb)~ 9 ' (4/) 



