Manchester Memoirs, VoL Lvui. {igig) No. At 7 



that must be changed to Oi = - + 6- Thus when is small the angle 



2 



0, of intersection approaches 90°, and Fig. i must be amended accord- 

 ingly. In this case, it is clear that of all the points A^, A2, . . , 

 A^\ A^^, . . , the nearest point to A will be A^^* for small values of 0- 

 Thus P^ consists of rows of transparent patches, these rows being 

 parallel to AAi\ i.e., making an angle ^i ( = 90° + Q) with X^. The 

 distance A2A^ ( =/i^) between adjacent patches in the same row is 

 clearly (oj + <^i) cosec Oi, whilst the distance between adjacent rows, 

 measured perpendicularly to them is />2^ = («2 + ^2) cosec Oi sin 

 e, = (a, + ^,). 



Thus ;^i^ = (^, + ^1) sec ^,)' /.v 



A'= a2 + i>2. / • • • • ^'""^ 



§8. The result Q^ of operation (iv) of §3 is easily seen to be equivalent 

 to rotating through 90^ the pattern P, described in §7. For Y^ is at 

 right angles to X„ whilst X, is at right angles to K,. We can combine 

 patterns P, and Q. following the method laid down in operation (v), §3, 

 but as is made small, there is no outstanding structure in the 

 resultant pattern, as was obtained in §6. For /i^ tends to the limit 

 ((7i+3i), as becomes indefinitely small, whilst /a' has the fixed value 

 (a, + (^2). Thus the combination tt, of P2 and Q2 is not coarser than 

 the coarser of the two original half-tone screens. 



§9. With regard to the pattern tti or network of squares described 

 in §6 and resulting from the combination of F^ and Q^, we note that 

 the axes of the pattern ttj, i.e., lines parallel to the sides of the squares, 

 'make angles f^/2 and (90° + Oj^) with the bands X^. The axes of the 

 pattern resulting from the combination of P2 and Q2, however, make 

 angles and (90° -f- O) with the bands X^. 



It has to be recognised that when the respective positions of the 

 two superposed screens S^ and S2 are given, each of the patterns 

 ^i) <2i> ^2, Q2 is exactly defined relatively to the others. Thus the 

 pattern ir^ resulting from Fi andi(2i is defined relatively to the pattern ttz 

 resulting from F2 and Q2. The effect of displacing the two screens 

 ^i and ^'2 relatively to one another, keeping constant is discussed 

 later. 



^10. The pattern F obtained by superposing two intaglio half-tone 

 screens, for which («2 + ^^2) / («i + '^i) is an integer, is now got by 

 combining the results of §§6 and 8, according to operation (v) of ^3. 

 It is now seen that when is small, the resulting pattern consists of a 

 very coarse framework of squares, each square being bounded by semi- 

 transparent bands or sides, whilst the interior is much darker and 



* In Fi^. I but with 6 (here our d^), made slightly greater than 90". 



