Manchester Memoirs^ Vol. Ixiii. (1919) No. 4 23 



y is made up of ////units (each of length yf\pt) ; then yi is made up 

 of ^/// units. Taking any pt. {i,=fjiy-^, r) = vy^) where fx and v are 

 integers, we see that the corresponding values of x and y in terms 

 of the unit referred to, are 



.T = ^ cos ^ - /; sin ^ = ^ - — ^ - = X {fxr- vs),\ 



f t J t f \ . {x) 



Thus a' and ji' are given by whole numbers. On subtracting multiple 

 of y (also whole numbers) to get the positions of the representative 

 points, we again get whole numbers. 



Thus the whole of the representative points will be on some or other 

 of the points of intersection of the series of squares given by 



pt pt 



where ^i^ and v^ can have any integral values, positive or zero, less 

 than pt\f. Associated with a square of side y there will be (^) 



such points. As there are (^) separate representative points, it 



follows that the points {xi) give all the representative points, in this 

 case (when/'^ and_/" are unity). 



By similar reasoning, we can show that when/' or/" is different from 

 unity, the squares (.r;) include all the representative points. Further, 

 whilst some of the points given by {xi) are fiot representative points, 

 points of the form* 



ffifii ff^f^'^ 



X = u" ■ 'J -^ . y V = ^'" - ^-^ -^ y, . . (xii) 



^ pt pt 



are representative points ; where /.t" and »'" are either zero or any 

 positive numbers less than /////'/"• Without being able to specify 

 more directly the positions of the representative points, we can assert 

 that they are not more widely spaced than the points given by (:v//). 



If now we remove the restrictions on ^', /ri", v^ and v", so that they 

 may now be any positive or negative integers (and zero), we easily see 

 that in the general case when O^ does not coincide with O^ that 

 the lines 



X = a + ^^^ , J- = « + "111. , . . . ixiii) 



pt ' pt 



* Notice that since/' is prime to/", pi Iff ^/'^^ is an integer, 



