Energy in the Electromagnetic Field. 11 



of the cylinder grazed by the rays are due solely to the 

 interference of these rays with those reflected from the 

 surface of the cylinder at varying angles. This would also 

 be the case as regards any plane such as QM' in advance of 

 the edge. But the phenomena in a plane such as Q'P', 

 which lies on the remote side of the edge, would not admit 

 of such simple treatment, especially when we consider the 

 effect at points lying not far from the boundary OY of the 

 direct and reflected rays. In such a plane, the intensity at 

 any point on the right of the boundary may be regarded as 

 due to the superposition of three factors : (a) the effect due 

 to the direct rays, (b) that due to the reflected rays, and 

 (c) a diffraction effect mainly perceptible in the neighbour- 

 hood of the boundary. If the cylinder were replaced by a 

 perfectly reflecting semi-infinite screen lying in the plane 

 CO with its edge at 0, the diffraction- effect would be found 

 as in Sommerfeld's * well-known investigation, by super- 

 posing upon the direct rays a radiation emitted by the edoe 

 of the screen. It will be observed that in the present case 

 the intensity of the rays regularly reflected from the surface 

 of the cylinder, as given by the formulae of Geometrical 

 Optics, is zero along the boundary OY, and increases slowly 

 as we move away from the boundary into the region of light, 

 and thus presents no discontinuity. It thus seems justifiable 

 to assume that so far as regards the phenomena in the region 

 on the right-hand side of the boundary, the diffraction-effect 

 (c) is practically the same as in the case of a semi-infinite 

 screen with its edge at 0. 



The distribution of intensity in the field may be readily 

 found on the foregoing assumptions. Take the " edge " 

 as origin of coordinates, and rectangular axes OX, OY', 

 perpendicular and parallel respectively to the direction of 

 the incident rays, and let the angle OCQ (assumed to be 

 small) be denoted by 6. The path difference between the 

 direct and the reflected rays reaching any point P is 



S = (QP_MP)27r/X + 7r 



= 2kd 2 {y + a&) + ir approximately, 



where k=2ir/\. 



Again, if the amplitude of the incident light be taken as 

 unity, that of the reflected light may be written as 



{pl( P + QP)}i, 



* Sommerfeld : " On the Math. Theory of Diffraction."* Math. Annalen,. 

 vol. xlvii. p. 317 (1895). 



