734 BELL SYSTEM TECHNICAL JOURNAL 



and of the wave-length X, travelHng along the x-direction in the 

 positive sense, shall fall normally upon it from behind. It is our 

 object to determine the amplitude of the waves in the region in front 

 of the grating. Suppose for instance that we select a plane parallel 

 to the grating and at the distance x in front of it, and derive the formula 

 for the amplitude at any and every point {x, y, z) of this plane. This 

 formula is the description of the theoretical diffraction-pattern in the 

 plane in question ; and the actual pattern may be observed by setting 

 up a screen or a photographic plate in the corresponding place. 



We went through this process for a single aperture in the first part 

 of this article; and there we found that the pattern is simpler (or, at 

 least, more calculable) the farther the plane of observation is removed 

 from the plane of the slit, being simplest when the two are infinitely 

 far apart. To realize this case in practice we have only to set a lens 

 immediately before the grating. Then, the diffraction-pattern appro- 

 priate to the plane at infinity — the so-called "Fraunhofer diffraction- 

 pattern" — is transposed into the focal plane of the lens, where we 

 must place the photographic plate in order to record it. Naturally 

 it is reduced in scale and augmented in intensity when it is thus trans- 

 posed, and for this as well as other reasons we had better express it, 

 not in terms of the coordinates {x, y, z) of the points in the focal plane, 

 but in terms of the direction-cosines (a = x/r, 13 = y/r, y = s/r) of 

 the lines drawn to these points from the origin of coordinates. ^ Our 

 formulae for the diffraction-pattern in the infinitely-distant plane are 

 in fact naturally expressed in terms of a, (3 and 7; and the lens may 

 be regarded as an agency whereby that value of amplitude, which 

 otherwise would have existed infinitely far away upon the line with 

 direction-cosines {a, j8, 7), is amplified by a constant factor and shifted 

 inward along this line to the point where it intersects the focal plane. 



We wish, then, to determine the vibration produced by a regular 

 sequence of slits, all over the plane which is either infinitely distant 

 or else the focal plane of the lens, according as the lens is absent or 

 present. 



Now we already have a formula for the vibration produced in that 

 plane by any slit individually. It is the formula (93) of the first part 

 of this article; to wit: 



5 = const. (1 + q;)[C sin (nt — mro) — S cos {nt — mrf^~\ 



, (1) 



= const. (1 -f a)-\C'^ + S^ sin (nt — w^o — e). 



' The origin should coincide both with the centre of the lens and with some 

 point in the plane of the diffracting apertures. This is impracticable; but the error 

 apparently does not make any trouble in practice, 



