Diffraction by a Semi-infinite Screen. 51 



deduced from the rigorous analysis for the case of the semi- 

 infinite screen has been made by Sommerfeld himself, and 

 also by Drude*, who has used the Cornu Spirals with good 

 effect in his discussion of the value of Sommerfeld's integrals. 

 According to these writers, KirchhofPs formula should give 

 the value of the intensity of illumination with sufficient 

 accuracy when the angle of diffraction is small, that is at all 

 points of the field (except very near the edge of the screen) 

 which are not far removed from the boundary between light 

 and shadow; but for large angles of diffraction, KirchhofFs 

 formula is inapplicable. It appears, however, from a 

 careful examination of the formulae given by Sommerfeld 

 and Drude, that the statement made by them on this point 

 requires to be qualified in one important respect. I propose 

 in the present paper to show by a detailed discussion that 

 when the screen is held very obliquely in the path of the incident 

 ivaves, the rigorous treatment gives results differing from 

 those of the approximate theory even in regard to small angles 

 of diffraction. Experimental work recently carried out by 

 me and described in the course of the paper confirms this, 

 and shows that the approximate theory of diffraction fails to 

 represent the facts correctly under these conditions. Inci- 

 dentally it is found that in the case of light polarized in a 

 plane perpendicular to that of incidence, the boundary con- 

 dition at the screen assumed by Sommerfeld leads to results 

 differing very widely from the observed optical behaviour of 

 any actual screen at very oblique incidences, and a suggestion 

 is made as to the manner in which the rigorous solution 

 should be modified in order to secure an agreement with the 

 result obtained experimentally in this case. 



Theory. 



It is convenient here to state Sommerfeld's results in the 

 simplified form obtained by him from a semi-convergent 

 expansion of t\\Q integrals representing the complete solution. 

 This is 



V2lT ~\ r2lT "1 



s=cos | — ?'cos(<£ — </>') + nt =Pcos — rcos (<p + (f) f )+nt 



1 /xc 1 l ~! riV 7t -i 



cos r —^ r - COS— -<p- J 



tan "^ ^ 



where s is the light disturbance, <f>' and <f> are respectively 



* Drude, < Theory of Optics,' English translation by Mann aDd 

 Millikan, p. 203. 



E 2 



