32 Application of Cornu's Spiral to Diffraction- Grating. 

 Substituting values of (f> and <^' we find 



'n'ir(a-{-h) sin ^\ 



sni 

 A = /. — 



. /7r(a + />) sin ^ 

 sm I — 



\ I 



the square o£ which is the usual expression for intensity. 



The discussion of the maxima and minima becomes ex- 

 ceedingly simple from the point of view of the above con- 

 struction. 



It will be observed that as (f> + (f>^ increases r varies back 

 and forth between the limits infinity and Z/2. 



The principal maxima will be when the chords / are all in 

 the same phase, and hence form a straight line of length nl ; 

 this can only occur when the radius r is infinitely great, that 

 is when (j>-\-(ji'=. ^irp or (a + h) sin = pX where p is any whole 

 number or zero. The corresponding intensities are ii^P. 



After passing such a maximum, as (^ + </)^ increases r becomes 

 smaller until the curve subtended by the n chords reaches just 

 completely around the circle. The intensity is then zero, and 

 the arc n (cj) + cp^) equals 27rnp + 27r, or differs by 27r from that 

 corresponding to the maximum. 



As 04-^ still farther increases r continues to decrease till 

 it reaches a value where the curve of chords reaches one and 

 a half times around the circle, or where n {(f> + <j>') = 27rnp + Stt. 

 This is veri/ nearly a subordinate maximum, though not exactly, 

 because, since r is decreasing, the maximum value will be the 

 chord of an arc a little less than Stt for which the radius is 

 greater. It is easily found by differentiation in the usual 

 way. 



Taking the approximate value n{(j) ■+■ (f>^) =37r, the amplitude 



being 2r becomes 



and as n is large we have 



H& 



that is, the first subordinate maximum has about -^2 ^^^ 

 intensity of the principal maximum. 



As r further decreases there will be a subordinate maximum 

 every time the curve of the n chords reaches around the circle 

 a whole number of times plus an additional half turn. Be- 

 tween two consecutive principal maxima there will thus be a 

 series of n — 2 subordinate maxima with absolute minima 



