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



It may be noticed that if / exists, it is the factor common to 

 q and t. If the G.C.F. of / and r be /', and of / and s be /", then 

 we can reduce the framework given by {vii). For if /' and /"be 

 other than unity, we shall have, on account of {iv), x increasing by a 

 multiple of y whenever m is increased by f^j/f^ and similarly y 

 increasing by a multiple of y whenever n is increased by p^ f//^^. 

 Thus within the framework of lines given by 



l = M^y,ptjff\ {M,^= ... -2, - I, O, I, 2, . . ), 1 ,... 



„ = iv;>,/////", (^/= ...-2, -I, o, 1, 2,..),/ • ^""'^ 



the pattern Ji exactly repeats itself in each rectangle. Further, this 

 framework is the least framework within which we have the whole of 

 the pattern R represented. 



It may also be observed that if the whole of the pattern R be divided 

 up into rectangles according to the scheme (wVV), we shall have for 

 each rectangle (Z"^)- //""/'/" points {x^y) of the general type ,„ On' 

 (/>., centres of black squares of S^) to be associated with each 

 rectangle. Boundary points can be settled by any convenient con- 

 vention. Corresponding to each of these points there will be a point 

 (a\ /30, where a} < y, ^' < y, satisfying 



x^o}^M^yA ,.. 



where M^ and iV' are integers. The points {a\ (i^) clearly be within 

 or on a square S given by x = o, x = y, y = o, y = y. These points may 

 be termed representative points for the point (o, /:>'). We see that the 

 representative points of those points ,nC?„', to be associated with each 

 rectangle given by {viii), will cover every possible representative 

 point for the given position of 6''. 



Representative points (a", /a'') possess the property that any displace- 

 ment of O^ from {x = a,y = ii) to (;v = a', j = /5'), keeping Q constant, 

 produces the same pattern R in another part of the plane. 



Further, of these {ptY\f~f'f'' points (a', /j'), no two (corresponding 

 to two points mi<^'nii mii^Sui) associated with the same rectangle of 

 the series {viii), can coincide. For if this were the case, a displace- 

 ment of 5, so that „uC>iii was changed to m„(9„,i (still keeping B constant) 

 would produce exactly the same pattern R in exactly the same place ; 

 so that the pattern A' would be periodic in a framework of rectangles 

 less than those given by {viii). This is, however, impossible. Thus 

 there are (//)' \f'f^f^^ different representative points in ii. 



We shall now discuss the representative points obtained when 

 (9' coincides with (9, i.e.., when n = o, /J = o. 



First, let us assume that the factors /' and /" are unity. In this 

 case we have (//)" //* representative points to consider. Assume that 



