CLASSICAL THEORY OF LIGHT 



737 



To prove this, and to find the magnitude of these equal steps, one 

 may proceed as follows. Omitting the lens again, consider in the 

 grating any two consecutive slits k and {k + 1), and on the very 



Fig. 2. 



distant screen two field-points P and P' separated by the same distance 

 c as separates corresponding points of the two slits — that is, the period 

 of the grating. Write down successively the formulae for the vibrations 

 produced by ^ at P and by (/^ + 1) at P'. They are, respectively: 



A sin (nt — niro — e^) ; A sin {nt — mro' — e^•+l). 



Since P lies in the same direction from k as P' from (k + 1), these two 

 are equal ; hence : 



ei+i - €i = m{ro' - Tq). (3) 



Here the factor {r^ — Tq) on the right is the difference between the 

 distances from the origin to P' and to P. In the limit when these 

 distances become infinitely great, all the lines from the origin and the 

 slits to P and P' become parallel and inclined to the plane of the 

 grating by the angle of which the cosine is /3; and the difference be- 

 tween the paths to P' and to P from the origin attains the limiting 

 value c/3. Hence in the limit: 



efc+i - ek = mc0. 



(4) 



This is the "step" or difference in phase between the contributions of 

 successive slits to the vibration at the field-point. The expression 

 looks more familiar if we put d for the angle between the normal to the 



