CLASSICAL THEORY OF LIGHT 735 



Here the symbol 5 stands for the ampHtude of the vibrating entity — 

 whatever that may be — at the various points ("field-points") where 

 the plane in question is intersected by the lines drawn from the origin 

 with direction-cosines (a, j3, 7); the symbols n and m for 2t times the 

 frequency and It over the wave-length of the vibration, respectively; 

 and the symbols C, S, To and e for various functions of (a, /3, 7). In 

 particular, C and 5 denote certain integrals extended over the slit, 

 so that they involve the breadth of the slit as well as the variables 

 (a, j8, 7) ; this last is true of e also; but ro denotes the distance from 

 the origin of coordinates to the field-point, and thus involves the 

 variables but not the breadth of the slit. As for the nature of the 

 vibrating entity which is designated by 5, I am keeping it intentionally 

 vague. Suffice it to say that 5 is something of which the phase cannot 

 be detected in any known way, but the amplitude controls the intensity 

 of the light; the observed intensity being, according to the classical 

 theory of light, proportional to the square of the amplitude.- 



If therefore we were studying the diffraction-pattern of a single slit, 

 we should be concerned only with the factor (1 + (x)^C~ + S~ in 

 the expression for s. It would be short work to develop the expres- 

 sions for C and S for a single rectangular aperture, finite or infinite in 

 length; and having developed them, we should have solved the 

 problem of the single slit; but in respect of our present purpose, it 

 would be a detour. Remarkable as it may seem, the pattern of the 

 single slit is only of secondary importance in determining that of a 

 regular sequence of slits. When we undertake to sum up expres- 

 sions such as (1) in order to compute the diffraction-pattern of such 

 a sequence, we find the emphasis violently shifted. A new set of 

 diffraction-maxima appear, and their positions are determined by the 

 variation of the phase {?it — mro — e) from one slit to the next — in 

 more general language, the variation which the phase undergoes in 

 passing over one complete period of the grating-structure. Mean- 

 while the influence of the coefificients C and S, and that of the breadth 

 of the slit which they involve, recede into the background. Not the 

 features of the individual slit, but the interval at which one follows 

 another, is now the dominant factor. This is the situation fore- 

 shadowed in the introductory pages. 



To bring this out, let us orient the s-axis in the plane of the grating 

 so that it runs parallel to the slits, which are of width a and are sepa- 

 rated by bars of width h so that the period c of the grating is equal to 

 {a -\- h). The A;-axis is to run, as heretofore, perpendicular to the 

 surface of the grating and through the centre of the lens, so that it 



^ Or rather to the sum of the squares of the amplitudes of several quantities, 

 any one of which separately satisfies the same equations as 5. 



47 



