Manchester Memoirs, Vol. Ixiii. (1919) No. 4 13 



whilst §17 shows us that the pattern will also follow (/3). It follows 

 that the resultant pattern must be of the " chess board " type. Assuming 

 that rtj is an integral multiple of a^, the pitch of the resulting pattern 

 as obtained from consideration (o) must be, from {vii), equal to 

 rti J 2 j 2sin (^/2). A similar result would hold from consideration (/3). 

 But this pitch for small values of $ is measured parallel to the diagonal 

 of a square. Thus the pitch of the resultant " chess board " pattern, 

 when measured in the ordinary way, viz., parallel to a side «„ is 

 fli / 2sin (0I2), and increases indefinitely as $ tends to zero. This result 

 should be compared with equation (vii). 



The effect of superposing two chess board screens with equal pitches 

 is shown in J^i'gs. ya and 7^ (F/. I//.)* 



The case of a^ja^ not being integral (a2>ai) can be dealt with on 

 lines exactly similar to those followed in ^$ 1 1 and 18. It is easily 

 seen that generally the side of the " chess board " pattern square plays 

 the part of {a^ + <^i) in §§ 4 to 18 



§20. We can extend the results of §5 to the case where 



ai + d^ = kd, a^ + d2 = /d . . . . (xvti) 



k and / being integers prime to each other. The appropriate diagram 

 for pitches / is shown in Fig. 8, which shows the points A of Fig. i 

 relabelled. Referring to Fig. 8, we shall have 



i/^i 2^1 = 2^1 3^1 = etc. = (<?, + /^2) cosec B, 

 ^A, ,.4, = ^A^ ^A. = etc. = (a, +l\) cosec Q. 



Now instead of grouping the transparent patches of which the A's are 

 corners, according to rows like ^Aj ^A^ ^A^ (say), we can group them 

 in rows parallel to ^A^ \^Ai. It at once follows from equations (xvii) 

 that i^i i^i^i^^i kAi,= k/d cosec. 0. If we regard the four points 

 i^i, iA\, \^A\, \,A„ as typical of a pattern of transparent squares, we 

 see that the corresponding pitches are 



Pz = i^i k^i = 2kid cosec sin {0J2) = ^/d sec (0I2) . (xviii) 

 p2 = kA\ N = kid cosec cos {0I2) = hkld cosec {Oj^) . {xix) 



It should be noticed, however, that we have rows of transparent spaces 

 having directions parallel to LM, passing through every point A on the 

 boundary of, or inside, the parallelogram ^At, ^A\ \JA\ \^A^. Thus, 

 calling the results just obtained pattern F^ (as in §3), we shall find on 

 superposing F^ and Q^f (according to operation (v) of §3), a whole 

 series of parallel square networks, all having their sides of square given 



* Owing to difficulties of reproduction, the two screens illustrated are not 

 exactly chess-board types, but consist of black round dots on a white ground. 



t<2 is, of course P^ rotated through 90'. 



