478 Prof. R. W. Wood on a Simple Treatment 



broaden as the slit- width decreases. These he called spectra 

 o£ the first class. In the case o£ the gratings used for optical 

 purposes, the lines are so fine that the central maximum of 

 the first-class spectra occupies the entire field ; i. e. there are 

 no minima, a single line scattering light of decreasing 

 intensity throughout the entire range between 0° and 90°. 

 In the present treatment we shall consider our lines of this 

 degree of fineness. " Absent spectra," resulting from finite 

 width of the line, and the consequent existence of first-class 

 minima, can be separately dealt with. 



We will consider parallel rays incident normally upon the 

 grating, the parallel diffracted rays being brought to a focus 

 by a lens. 



Each line of the grating acting alone, we will suppose to 

 produce unit amplitude at the focus. 



We find the resultant amplitude produced by a number of 

 lines operating together by the well-known device employed 

 in the elementary treatment of Cornu's spiral, the resultant 

 amplitude being the closing side of a polygon, the sides of 

 which (vectors) represent the amplitudes and phases of the 

 vibrations coming from the grating-lines. We can plot the 

 intensity curve for a three-line grating, by considering phase 

 differences (P.D.) which increase by 20°. In the normal 

 direction (P.D. 0°) the intensity will be 3 2 or 9 ; in a direction 

 such that we have a P.D. of 90° the intensity will be 1, 

 while with a P.D. of 120° we have a triangle, tbere is no 

 " closing side " and the intensity is zero. From now on it 

 increases, attaining the value 1 again with a P.D. of 180° 





Fig. 1. 





b R D - 0° 





*'' RO =¥f 



a" 



RD =90° 



A 



RD ^\ZQ° 



4^ 



RD -135° 



< — » 



/?D =180° 



when the three vectors are superposed ; two of these vectors 

 cancel each other, the illumination being that due to the out- 

 standing one. The various stages are shown in fig. 1 for 





