4 Lees, On the Superposing of Tivo Cross-line Screens 



of the type ABCD. We shall take the set of bands parallel to AB as 

 being the set A",, whilst bands parallel to BC constitute the set A',. 

 Let the width of a transparent band of A'l be a^ and the width of a dark 

 band of X^ be b^. Let a^ and b., represent the corresponding widths 

 for set X^. It is easily seen that 



AB — a-, cosec Q, BC =«, cosec Q, ^ l\ 



A,A = B,B = {a, + b.) cosec Q, AA,' = BB,' == {a, + b,) cosec e.j ' ^'' 



The pattern B^ of transparent parallelograms of the type ABCD may 

 be considered as built up of transparent patches arranged in a series of 

 parallel rows. Thus the patches may be regarded as arranged in rows 

 parallel to AA.^, or to AA^, or to AA., etc. The natural grouping of 

 these patches for small values of S will be optically that corresponding 

 to rows parallel to ^^n+i (say), where the point ^n+i has the least 

 distance from A, of all the points A^, A^, etc. Since the distance 

 A,An+i = nA,A2 = n {a, + b^) cosec 0, we find that 



AA n + i I 



= cosec^e[{a2 + b^Y + n^(a, + b,y — 2fi{a^ + b^) (a, + b,)cos 0]> . (ii) 

 = E cosec^O (say). / 



For a given value of ^, ^ is a maximum when 



n = {a^ + b.^) cos e I {a,+b,). . . . {Hi) 



Assuming that (a^ + b^) is not less than (a^ + bj^), we shall therefore get 

 the best grouping, generally speaking, when the integral number n 

 chosen is that nearest to (a^ + b^) cos / (a^+bi). As most of the 

 results we wish to investigate immediately become prominent when 

 is very small, it follows that, in such cases, we take for n the nearest 

 integer to (a^ + A,) / (a, + b^). Thus if a^ + b^ = ai + b^, we should take 

 the transparent patches in rows parallel to AA,. 



To simplify the discussion, we shall now assume that {a^ + b,) is a 

 multiple of {a^ + ^i), and we can conveniently denote the multiple by 

 n. Li this case, for small values of $, the rows of patches are parallel 

 to AAnJ^z- It is worth enquiring for what values of this statement 

 will hold. 



Since AA\,+, = cosec^^ [n^ + ii" - 2«^ cos B\ («, 4- b,Y \ ,.. 



s,nd A A-'a^ cos&c-e[ff~ + {n- iY — 2n{n-i) cos ^]{a^+b,Y } ^ 



it follows that AA„^,< A A,,, \( cos $ >{2n - i) j 2«,) / . 



;>., if sin 6* < (4"- 0''"7 2«- ) ' ' "< ' 



Thus when « is large, the range of values of 9 for which we can take 

 the same row of transparent patches throughout {viz., AAn+,) becomes 

 small. 



