8 Lees, On the Superposing of Two Cross -line Screens 



consists of a sort of granulated structure as outlined in .^8. Further, 

 the smaller the angle 9 is, the coarser the square framework becomes, 

 and there is no limit to the size of the squares as diminishes 

 indefinitely. These results are illustrated in Figs. 3 and 4 (PI. I.), 

 which exemplify clearly the above analysis. 



It should be borne in mind that if {a^ + 1?^) / (a, + b,) is not an integer 

 it is impossible to get the square framework of R of indefinitely great 

 coarseness by making become indefinitely small. We shall proceed 

 to show this. 



.^11. We shall assume that a^ + l?^ = {n + e) {a^ + b,)* where t is a 

 positive or negative irrational fraction numerically iless than o"5, and n 

 is an integer. For simplicity, we shall write 



ai + bi=d^; a^ + b^^d^ . ■ . . {x) 



so that d2 = d^{fi + f) . .... (xi) 



In this equation, n is to be interpreted as the nearest integer to the 

 fraction d^jd^, which is of course to be taken as having a value greater 

 than unity. We can still use Fig. 2, and with the value of n just taken, 

 the transparent patches must still be considered (for small values of fj) 

 as arranged in rows like AAn+-i. Since now AA^ is not exactly equal 

 to A^An+i, it follows that the direction AA,^^^ no longer bisects the 

 angle A^AAi\ It is interesting to trace the variation of the angle 

 A,AA,' ( = <^ say). Although AA, and AA,' are still given by (/), and 

 thus vary with 0, yet their ratio remains continually the same. Thus 

 for direction purposes, the lengths of the sides in Fig. 2 may be con- 

 sidered as constant. Regarding A^A as a fixed line, the point A^^^ 

 will lie on a circle having A^ as centre. If t be a positive fraction, the 

 point A will lie outside this circle and the greatest value that 4^ can 

 have will be attained when the line AA^+i touches the circle having 

 centre A, and radius yi,/^,i+i- This will clearly be the case when 



,, AA,' n , ..V 



cos^=— -^= .... lyXll) 



AA^ 71 + t 



If we denote this value of by 0^, we see that the corresponding value 

 of ^ is 90° — ^,. We also see that as ^ diminishes down to ^i, ^ increases 

 up to 90° - Oi] SLS further diminishes, <p dimishes steadily, until when 

 = 0°, ^ also becomes 0°. When, however, £ is a negative fraction, 

 <■/» increases steadily as diminishes, and ultimately, when = 0° 

 f becomes 180°. 



We shall now find the value of the pitch /. (i.e., the perpendicular 

 distance between the lines AAn+^ and A^K). Denote by \) the length 



The case of £ a simple fraction is discussed in § 20. 



