Manchester Memoirs^ Vol. Ixiii. ( 1 9 1 9) No. 4 1 5 



by {xix). Each of these square networks will be of the type described 

 in §6, but it is important to note that there will be kl distinct networks 

 of this type, when account is taken of all the points A. These 

 networks of squares of side /a can be again regarded as consisting of 

 tivo parallel square networks only One network will have a side/a/'^j 

 the other a side /,//. It is clear, however, that on account of the size 

 of /2, with large values of k and /, it will be difficult for the eye to pick 

 out the regularity of the pattern ; more particularly will this difficulty be 

 emphasised on account of the presence of two sets of parallel networks 

 of pitches ///^ and /// respectively. It must be emphasised, however, 

 that the resultant pattern of P.^ and Q^ (according to operation iv) of §3) 

 is periodic within squares of side />,, and this can be shown by making 

 ^ very small, when the periodicity becomes recognisable. 



The effects of adding patterns P., and Q^ to the pattern just 

 described, can be obtained by the procedure given in §§7 and 8, and 

 we have in no way to modify the above argument for the coarse 

 framework of R. 



The results given in this section can be obviously applied mutatis 

 mutandis to the case of superposing two ordinary half-tone screens. 

 Thus the two sets of square frameworks, consisting of dark boundaries 

 with lighter interiors, will have pitches or sides given by p^lk and /,// 

 respectively ; where p.. is given by equation {xix). 



%2i. We shall now briefly discuss the average intensity of light after 

 transmission through two ordinary half-tone screens [i.e., 5",' and .S^' 

 of §13). The results here given must be taken as those averaged over 

 large areas of each plate or screen. 



Assuming the uniform intensity of the incident light to be /, the 

 average intensity after passing through a screen ruled only with bands 

 X^"- (§13) will clearly be I.b^ j {a^ +^i). Superposing two such screens 

 at 90" gives us the effect of S^^ Thus after transmission through the 

 half-tone screen Si\ the intensity of the transmitted light is 



I,^^/i>rl(a, + d,Y (xx) 



If the light had been passed instead through 6*2', the transmitted 

 intensity would have been 



I^'=I K^'lia. + hY {xxi) 



The final intensity, after passing through both screen S^,"^ and 6",', 

 is therefore 



I.' = I b^^b^"" l{a, + lf,Y{a^ + b^Y. . . (xxii) 



Similarly, if S^ and ^'2 be the negatives of 6V and ^'2^ the respective 

 intensities of transmission are 



(for S, alone), /^ = /—/,' = /. {2a,b, + a,y j (a^ + b,)'-. . (xxiii) 



