24 Lees, Oji the Superposing of Two Cross -line Screens 



give a network of squares, the corners of which (when inside the square 

 2 of side y) include all the representative points of the point (a, /3). 

 The points given by 



pi ' pi 



provided they lie in (or on) il, are all representative points. 



Equations {xiii) define {pt\fY points, whilst {xiv) define {pt\ff^f^^Y 

 of the representative points, the total number of which is {ptYlf^f f^^- 



Considering then, only displacements of O'^ inside the square S, we 

 see that if the pattern R is to remain unaltered in design, O' must 

 always coincide with a representative point of the original position 

 («, /i) of 0\ 



Equations {iv) show us that we still get periodicity of the pattern R 

 within a rectangle, provided g cos 0\p and q sin 0\p are commensurable 

 {e.g., glp= J 5> co^ ^ 2 j J S' sin 0= i j J 5 ). Whenever this 

 condition is not satisfied, there is no periodicity in the pattern, i.e., no 

 finite displacement of S^ over S will produce the same pattern R in 

 the same place. Thus any displacement of 0\ from one point in such 

 a square to another point in the same square, will produce a pattern R 

 totally different (i.e., it cannot be made to coincide with the original 

 pattern by a finite displacement of R). 



In such cases as these (and they are the most general) it is strictly 

 impossible to talk of the pattern /? as being defined by 6* alone ; the 

 relative co-ordinates of (9' to O should also be specified. 



Disregarding all cases when {) = o, we can examine some simple 

 cases where periodicity in pattern R exists. The coarsest set of 

 representative points is obtained when 



(these values of r and s can be interchanged). In this case, from 

 equation (iv), we see that the pattern R is periodic in a square frame- 

 work of size yi. Thus for a given position of 6>' within a square 

 of side y, there is no other position for O' inside the square, which will 

 reproduce R in any part of the plane. Similarly when 



qlp^^-JmJ^V^ — = — ; f'h and n, being integers. The next 



simplest case is where 



