Manchester Memoirs, Vol. Ixiii. (191 9) No. 4 21 



If a point P have co-ordinates (£, t]) relative to 0% O'r], then its 

 co-ordinates {x,y) relative to Ox, Oy will clearly be given by 



:x; = a-1-^ cos ^ — j; sin ^, \ ,... 



y = /3 + ^sm d+ncos e,) ' ' ^"' 



If the point P coincide with some point mOn, which is the centre of 

 another black square of S\ and which has co-ordinates ^ = w/i, rj = ;/y„ 

 where m and n are integers, then 



x = a + yi (m cos 0~fi sin 6),} /•.•\ 



j>^ = /? + yi (''''^ sin 6* + ^^ cos ^).j ' " ' ^^^^^ 



We shall assume that y is a commensurable fraction //^ (where 

 p and (7 are integers) of y^ ; also that cos and sin are commen- 

 surable, and equal respectively to rjt, sjf, where r, s and / are integers. 

 We can then write 



X = a + -^ (mr - fis) = a + -^ yimr - ?is), 



y = 13 + — {'ns + nr) = (3 + -^ y{ms + fir), 

 t pt 



It is, of course, assumed that the fractions p\q, rjt, sjf are expressed 

 in their lowest terms. Since 



{rltY + {s\tY=i . . . . (v) 



we may take it from the theory of numbers, that / is always odd, 

 whilst one of r and s is odd, the other even. Further ;' is prime to s. 



We see from equations (w) that if m and n are multiples of pt (or if 

 a common factor / is present in the numerator and denominator of 

 (/ Ipt, we can take m and n multiples of////), then we can write 



X = a + My \ 



y = (3 + Ny j ■ ■, ■ ■ ^^ 

 where M and 2V are integers. 



Thus relatively to a new origin having co-ordinates x = My, y — Ny, 

 we find that x = a, y = f3, exactly as in Fi^. 9. From equations (I'v) 

 we see further that every time either m or n is increased by pt\f, 

 i; and ?/ are both increased by multiples o^ pty^ j/,. and we get new 

 values of J/ and JV satisfying (vi). 'i'hus we have established the fact 

 that with a square framework of lines defined by 



S, = M.y.ptjf, {M,== ... -2,-1, o, I, 2, etc.), \ 



y] = N^y^pt\f, {N^= ... -2,- I, O, I, 2, etc), j 



{vii) 



the pattern R exactly repeats itself in each square, independently of 

 (a, j8), and we need therefore only consider one square. 



