542 Transactions of the Society. 



Our first step must be to determine the diffraction-spectra 

 produced by such structures. We know that a grating of simple, 

 straight, and narrow slits like fig. 123 gives a row of diffraction- 

 spectra lying at right angles to the direction of the slits. Suppos- 

 ing we place another simple grating across this one in the manner 

 shown in fig. 124, the result will be that the light from those parts 

 of each original slit which are covered by the bars of the second 

 grating is cut off; but the light from the portions of the original 

 slits which remain uncovered necessarily continues in the same 

 phase-relation as before, and therefore produces precisely the same 

 row of spectra, only proportionately weakened in brightness. 

 Hence a row of bright dots produces essentially the same diffraction- 

 spectra as the unbroken slit, of which the dots may be considered 

 to be intermittent portions. But this deduction immediately leads 

 us to another; for if the dots are arranged in any perfectly 

 regular order, they will range themselves into straight rows in a 

 number of different ways and directions, and thus we are justified 

 in laying it down that a dot-pattern produces rows of diffraction- 

 spectra corresponding to all simple line-gratings, the slits of which 

 have a direction in which the dots form themselves into straight 

 rows. "We will study two concrete cases to make this abstract 

 proposition clearer. Let us first take bright spots (or perforations) 

 arranged in perfect squares (fig. 125). We can range these — 



1. Into horizontal rows a, a..., corresponding to a verticaL 

 row of diffraction-spectra A u A 2 , etc., in fig. 125a. 



2. Into vertical rows b, b . . . , corresponding to a horizontal 

 row of diffraction-spectra B x , B 2 , etc., in fig. 125a. 



3. Into two oblique rows, c, c and d, d. . ., with corresponding 

 rows Ci, C 2 . . ., Dj, D 2 ..., of diffraction-spectra; and we note 

 that the lines c, c and d, d are closer together, hence the corre- 

 sponding diffraction-spectra are further apart. 



4. We can arrange those dots into rows which are in the 

 relation of a knight's move on a chessboard — with four possible 

 directions e, /, g and h ; the distance between these rows will be 

 still smaller than that found in case (3), and the diffraction-spectra 

 E, F, G, H, will be correspondingly further apart. 



Evidently this may be carried further and further ; the principle 

 will, however, now be perfectly clear. 



The result is that the dot pattern here considered gives a set of 

 diffraction-spectra precisely similar in arrangement to the pattern 

 itself; for a simple mathematical investigation shows that the 

 increasing closeness of the oblique rows of successive orders is 

 such as to cause the corresponding diffraction-spectra to be spread 

 out to the proper distances to cover the right places in our pattern. 



It should be pointed out that both fig. 125 and fig. 126 show 

 the dots black instead of white, after the manner of a photographic- 

 negative. 



