744 BELL SYSTEM TECHNICAL JOURNAL 



non-existent, evidently the maxima in question are blotted out. This 

 will happen, for example, to every maximum of even order, if bars and 

 slits are equally wide. For, taking the direction d^ism d^ = 2X/c) as 

 an instance : the contribution made to the total vibration by the upper 

 half of each slit will be equal in magnitude and opposite in phase to 

 that made by the lower half, and the total contribution of the slit will 

 be zero. If in the spectrum produced by a grating the even orders are 

 missing, or if — to say what would actually be noticed — the values of 

 sin 6 for the present maxima stand in the ratios 1 : 3 : 5 : 7 • • • 

 instead of 1 : 2 : 3 : 4 • • • , the inference is that the grating has been 

 so ruled that over half of every period the phase of the emerging 

 (transmitted or reflected) light is constant, and over the other half 

 no light comes forth at all; as for instance would be the case if half 

 of every period were the unmarred surface of the metal, and the 

 diamond had made the other half perfectly black. The reader may 

 work out for himself what it must mean if every third, or every 

 fourth, or every wth of the maxima is absent. 



We return now to the expression (equation 2) for the contribution 

 of a single slit or period of the grating and rewrite it, taking due account 

 of our subsequently-gained knowledge that ^a; is constant and (pk 

 increases by equal steps mc sin 6 — mcl^ from slit to slit: 



Sh = const. (1 + a) /I sin (nt — mro — eo — kmc^). (9) 



For convenience number the slits from to TV — 1, representing by 

 N their total number (formerly called 27lf, but now there is no reason 

 for supposing it even), and locate the origin so that mro = eo- Gather- 

 ing all the factors of the sine-function under a single symbol B, and 

 writing out the expression for the summation of Sk from fe = to 

 k = (N — 1), we find for the resultant vibration in the direction 6: 



N -1 



5 = 5 X^ sin (nt — kmc0) 

 t = o 



= B sin w/(l + cos a -\- cos la -\- • • • cos {N — \)a) 



(10) 

 — B cos «/(sin a + sin 2a + • • • sin {N — \)a) 



= B'^c sin nt — BJ2s cos nt, 



in which a stands for wr/? and Xl-- ^^^(^ L* for the finite series of 

 cosines and sines which are indicated. 



For the amplitude of the vibration — the only thing which matters — 

 we then have 



D - 5Ve<- + e;'- (10 



