and the Dynamical Theory of Diffraction. 437 



The following constructions can also be used for obtain- 

 ing the direction of the electric and magnetic quantities. 

 Draw a sphere around the orifice and draw an axis through 

 the sphere in the direction of the incident light. Then rule a 

 sheet of paper with a series of lines at right angles to each 

 other. Cut a star-shaped piece out of the paper with its dia- 

 meter equal to the circumference of the sphere, and having a 

 very large number of points. Place the centre of the star 

 on the sphere at the end of the axis where the light leaves 

 the sphere, and wrap the points around the sphere, the points 

 meeting around the incident ray. The marks on the paper 

 then give the required directions. 



We can also construct the curves of polarization by noting 

 that the stereographic projection of the lines on a plane is 

 merely a series of straight lines. 



The equations become very simple at many wave-lengths' 

 distance from the orifice, especially wdien the orifice is small. 

 If the radius of the original wave is large, it is usually suffi- 

 cient to consider the periodic factor as the only variable. In 

 this case we can write 



0'=|^cos<£[l + cos0] ffe-^-v^ 



ibl 



i — ' 



^sin <£[! +cos 0] ffe-^-v*)^ 



P' = 0, 



where I is the coefficient of the original vibration, and there- 

 fore its square is the intensity of the original light. The 

 intensity of the refracted light is simply proportional to the 

 sum of the squares of ®' and <&'. This is the expression 

 ordinarily used except the term in 0. Thus for a circular 

 orifice we have the vibration expressed in Bessel's functions, 



^ ^ r j /i - m Ji(br sin 6) ..,„ VA 

 ©'=7715 cos <K 1 1- cos 6) — ^ — -= — '- € -«*(tt- vo 



2K rv ; suit/ 



•w ^ r • j /i , /i\ 3i(br sin 6) ,™ VA 



It is impossible to pass from these expressions to the case of 

 a plane wave, since they are only for the case of a great 

 distance from a small orifice. 



