738 BELL SYSTEM TECHNICAL JOURNAL 



grating and the direction from the grating to the field-point, so that 

 /3 = sin ; then : 



CA-t-i — f^- = I'^c ^in ^ ^ ~r^' ^'''' ^- (^) 



Thus we have arrived at the conclusion — indeed almost self-evident — 

 that the consecutive slits of the grating supply to the total vibration 

 at the field-point contributions which are exactly equal in magnitude 

 and follow one another at equal intervals of phase. 



Our problem therefore is to sum up the series of these contributions. 

 The process is an easy one; but we shall be able to foresee the major 

 feature of a diffraction-spectrum without even writing down the 

 summation. For it is evident that there must be maxima of vibration- 

 amplitude, maxima of diffracted intensity, at the field-points or in the 

 directions where the contributions of all of the slits agree in phase — 

 that is to say, differ in phase by integer multiples of lir. Counting 

 outward from the centre of the diffraction-pattern, or normal to the 

 grating, the first of these maxima must lie in the direction for which the 

 "step" in phase of equation (5) is equal to 27r; hence for this "first- 

 order maximum": 



sin e = X/c. (6) 



The second lies in the direction for which the step in phase is equal to 

 twice lir; the third in the direction for which the step is thrice lir; 

 and in general there is a sequence of maxima, the general formula for 

 the wth of which is the celebrated "plane-grating formula": 



sin dn = «X/c, w = 1, 2, 3, 4 • • •. (7) 



The symbol n stands customarily, and in this article henceforth shall 

 stand, for the order of the maximum; from now on I will write liru 

 for the quantity which before was denoted by n. 



These are the great principal maxima of the spectrum cast by a 

 diffraction-grating. There are others between, but in practice they 

 are inconspicuous or invisible. Thovse of which I have just derived the 

 locating formula are the maxima from which wave-lengths are com- 

 puted. Let it be emphasized again that the formula was derived 

 without taking into account the ratio of slit-width to bar-width, and 

 that it does not involve the width of the individual opening, but only 

 the spacing between corresponding points of consecutive slits. In- 

 deed, if one examines the deduction, it will be seen that really nothing 

 peculiar to a slit enters into it at all. All that is preassumed is that 

 the grating sends to the field-point a series of component vibrations, 

 ecjual in amplitude and stepped off eciually in phase. Such is indeed 



