of Problems on the Diffraction of Light. 81 



elements down to the first, it must also be true for the first, 

 because the whole resultant shows a difference of phase cor- 

 responding to that distance. In the usual way of treating 



the subject, PMj is taken to be PO + ~ ; the result being that 



the first element produces a phase differing considerably from 

 that of the succeeding ones : by reducing the width of the 

 first strip we make the difference of phase nearly equal to that 

 of the others. 



The following Table shows the phases of the vibration at P 

 due to successive strips, as calculated by means of Fresnel's 

 integral, both for the case in which the old division is taken 

 and the one here suggested. 





Usual division 

 into strips. 



Suggested division 

 into strips. 



Eesultant Phase due to first strip 



53° 20' 



42° 40' 



,, „ second strip... 



180°+80° 18' 



180° +38° 15' 



,, M third strip ... 



86° 15' 



44° 20' 



Converging towards 



90° 



45° 



and 



180° +90° 



180°+45° 





If, then, the usual division into strips is adopted, and the 

 amplitude due to all the strips is added up algebraically, as 

 has been done by Mascart and others, a difference of phase of 

 37 degrees is treated as non-existent. On the other hand, in 

 the division which I suggest, all phases are nearly equal. The 

 second strip shows the greatest difference of phase, but as the 

 cosine of 7° differs by less than one per cent, from unity the 

 results of the calculation will be sufficiently accurate. 



We shall apply this new division into strips to calculate 



the position and intensity of diffraction-fringes at the edge of 



shadows cast by straight boundaries. It will be necessary in 



the first place to calculate the amplitude produced by each 



strip separately. Assuming, as before, that the phase of 



vibration at P produced by the subdivisions of each strip 



after the two central ones, range uniformly over two right 



2 A 

 angles, the amplitude produced at P will be - 



IT 



where h is the width of the strip. Neglecting squares 

 of \, we find that the width MiM 2 of the second strip is 



