Manchester Memoirs, Vol.lxiii. (19 19) No. 4 25 



Here again r and s can be interchanged. In this case, f=f^ =/" = ij 

 and equations {^xiv) show that the representative points for a given 

 position (a, ^) of O^ within the square of side y are given by 



a; = ct + /i"— , j = a+)'"-^, . . . iyxvii) 



where /.i" and v" are integers chosen so as to malce the points {xy) lie 

 in or on the square of side 7. Thus the side of the squares defined 

 by {xvii) is of length y / 5. 



In all other cases, the distribution of representative points will be 

 finer than that given by {xvii). 



These results show that where periodicity of the pattern R (or ^') 

 exists, the representative points will be so finely distributed, except for 

 the cases just mentioned, that we may for practical purposes discuss 

 the pattern as depending only on ^. 



The most general case is that in which either y^ \ y, or cos B and 

 sin Q (or all of these) are incommensurable in such a way that there is 

 no periodicity in the pattern R (or i?'). In this case for a given (a, /?) 

 there will be a doubly infinite series of representative points (corre- 

 sponding to the doubly infinite series of points mPn). We shall 

 not attempt to discuss the distribution of these points, beyond the 

 following : — 



(/) To any point (o, /?) inside the square of side y there will be a 

 doubly infinite number of representative points (a', ^^) inside 

 the square. 



(;V) A displacement of (a, /j)') to any of these points (o^, /^O will give 

 exactly the same doubly infinite series of representative points, 

 but of course the points mO^ to which these correspond, 

 will have changed. 



(///) If (a', /5') be a representative point of («, /i), then the point 

 («",/3"), where „ii^ («'-«) ^ ^n = itA^ , N and N' 



being integers, is also a representative point of («, /5). Thus 

 the representative points inside the square of side y are 

 infinitely crowded everywhere. 



We are thus entitled to say that although we can never get, in such 

 a case, exactly the same pattern R on displacing (9' from (a, ^) to 

 («S^')) unless (a', /30 is a representative point of (a, /3) ; yet we can 

 always find some part of the plane where the new pattern R will differ 



