736 BELL SYSTEM TECHNICAL JOURNAL 



intersects the focal plane (or the plane at infinity) at the point which 

 is the centre of the diffraction-pattern. The 3'-axis is to lie in the plane 

 of the grating perpendicular to the slits. The light comes up to the 

 grating normally from behind, and therefore follows the x-direction. 

 For reasons which will presently appear, it will suffice to calculate the 

 diffraction-pattern over not the entire focal plane, but only the line 

 where this is intersect ed by t he xy-p\ane. For any field-point on this 

 line 7 = and a = ^ji — ^'. 



To the total vibration at any field-point, each slit now makes a 

 contribution given by the expression (1). Numbering them in order, 

 we may write for the contribution of the ^th slit: 



Sk = const. (1 + a)Ak sin (pk, (2) 



in which Ak stands for the value of VC^ + S-, and (pk for the value of 

 (nt — mro — c), appropriate to the ^th slit. 



Now Ak has the same value for all the slits. This may be proved 

 directly from the formulae ^ for C and S, or indirectly by the following 

 chain of reasoning. The function VC- -f 5^ describes the diffraction- 

 pattern formed by the single slit on an infinitely-distant screen, when 

 there is no lens. Two similar slits a finite distance apart would 

 produce two such patterns, one displaced by the same finite amount 

 relatively to the other. But on the infinitely-distant screen the fringes 

 and other details of the patterns are themselves infinitely broad, so 

 that a finite displacement of one with respect to the other leaves them 

 still practically — and, in the limit, exactly — in coincidence. This 

 remains true when the patterns are transposed to the focal plane of 

 the lens; those produced by a slit in one place coincide exactly with 

 those which would be produced by an exactly similar slit lying any- 

 where else.^ Therefore VC^ + S~ must be the same function of 

 (a, /3, 7) for every slit. 



At first glance this argument seems to prove that the diffraction- 

 pattern for the grating is merely that of the individual slit, multiplied 

 manyfold; but that conclusion would in general be false, for we have 

 not to add amplitudes but to compound vibrations with due regard to 

 their relative pheises. The phase (pk which figures in equation (2) 

 differs from slit to slit; and if these follow one another at equal 

 intervals, (pk changes from one to the next in equal steps. 



' By operating on the expressions (presently to be derived) for Cand 5 in the case 

 of a rectangular a])irtin-e, one may show that, while each separately varies when the 

 position of the rectangle with reference to the origin is changed, the sum of their 

 squares remains the same. As any finite aperture may be regarded as a collection 

 of finite or infinitesimal rectangles, the theorem is general. I am indebted to Mr. 

 L. A. MacColl for working this out. 



^ The practical limitation to this statement would be set by the impossibility of 

 making an ideally perfect lens of indefinitely great size. 



