1 8 Lees, On the Superposing of Tzvo Cross-line Screens 



constant, is clearly equivalent to sliding Fig. 12 over Fig. 11, still 

 keeping bands Kj in both figures in coincidence. 



Either of the slidings of Fig. 10 over Fig. 9, or of Fig. 12 over 

 Fig. II, may be zero in special cases of the relative displacements 

 of S^ and S^^. 



We also see that when the corners of the parallelograms shown in 

 Figs. 9, 10 and 11 are fixed, owing to the mutual positions of S^ and S,, 

 being absolutely specified, then also are the corners of the parallelo- 

 grams of Fig. 1 2 specified. Thus the positions of the centres of the 

 transparent spaces of Qr,, are defined, when those of the centres of the 

 transparent spaces of P^, P^ and (2, are given. In other words, the 

 spacing of the pattern Q^ is dependent on the spacings of the patterns 

 P„ /'a, and (2i. More generally, the spacing of any one of these four 

 patterns is determined uniquely by those of the other three. 



However, if it is recognised that for small values of Q., the pitches 

 // and p:^ of equation (ix) are small compared with p,,., it will be seen 

 that the displacements referred to will not materially modify the main 

 or coarse framework of squares, and this is readily verified by 

 experiment (see also analytical discussion in Appendix). 



§23. Conclusio?i. The author has discussed in this paper the 

 general characteristics of the patterns obtained on superposing two 

 half-tone plates of like type, at small angles. More particularly, the 

 cases of (/) intaglio, (//) ordinary half tone, and {Hi) " chess board " 

 screens have been discussed. In these cases, it is shown that the 

 coarse square framework which arises is similar in type to that of each 

 of the constituent screens. Formulae for the pitches of the coarse 

 square frameworks are deduced, and it is shown that if the pitches of 

 the original screens are to one another in the ratio of simple whole 

 numbers, a coarse square framework arises, whose pitch varies as 

 cosec {d\2), and thus become infinite when $ tends to zero. In other 

 cases, it is shown that the pitch remains finite always. 



It has to be remembered that all kinds of groupings of the 

 transparent spaces forming the final pattern are possible tlieoretically, 

 but the ones taken for consideration are those that force themselves on 

 the eye. 



An important point brought out in the paper is that the resultant 

 pattern R of two intaglio screens S^ and ^2 is not the negative of the 

 pattern R' which is the result of superposing screens .S",' and S^} which 

 arc the negatives of 6", and S., respectively. 



The important cases which arise when is large are not discussed 

 in this pajjcr. The questions of framing rules f(;r olitaining the best 



