CLASSICAL THEORY OF LIGHT 745 



Now, as may easily be proved ^ : 



VEo- + L.- = sin i^Na) : sin (^a) (12) 



so for the amplitude of the vibration in the direction 9 we have: 

 „ sin {^Nmc sin 6) . 



U = D —. jr^ • „. ■ • (13) 



sm {^nic sm 6) 



Here we have that product of two factors which was foreshadowed in 

 the early pages of this article — one factor (the second) depending on 

 the periodicity of the grating, and controlling the location of the 

 diffraction-maxima; the other depending on the structure of the 

 individual slit or groove or atom-row, and controlling their intensity. 



The second factor displays the qualities which have already been 

 deduced by simpler means, and others. It vanishes whenever ^Na is 

 an integer multiple of x, except when simultaneously |a is an integer 

 multiple of tt, in which exceptional cases the great principal maxima 

 occur. These are not the only maxima, for between any two of them 

 there are {N — 1) equally-spaced minima (directions where \Na is an 

 integer multiple of tt but \a is not) and between these in turn there are 

 {N — 2) maxima of which the locations may be found by the usual 

 method. These so-called "secondary" maxima are however faint and 

 inconspicuous, having, according to Wood, but 1/23 the intensity of 

 the principal peaks, unless the grating is composed of only half-a- 

 dozen lines or fewer. 



The first factor consists essentially of that function 



(1 + a)iO + S' 



mentioned in equation (1) and earlier, which describes the diffraction- 

 pattern of the single slit (or groove, or atom-row). Wherever that 

 diffraction-pattern has a zero of intensity — the "centre of a black 

 fringe," to use the common language — the intensity in the pattern of 

 the grating is likewise forced to vanish. W'hen the slit occupies half 



' One method is based on the fact that 2c and i^s are respectively the real and 

 imaginary parts of 2e'*", so that 



Further, by a well-known formula 



^ eika = 1 -|- ga-a j^ (e'''")- -f • • • (e'*a)iV-l 

 



= (1 - et.va)/(i - e^") 



and there is a corresponding expression for e'^^", multiplying the two of which together 

 and taking the square root one arrives directly at the stated result. 



