[573 
573] NOTE ON THE (2, 2) CORRESPONDENCE OF TWO VARIABLES. 95 
or, what is the same thing, we have 
1 :-(0 + x) ■ 8 X = {a,...\<f>, 1, 0)» : 2 (a, 1, Op, 0, 1) : (a,...p, <f>, l) 2 
= acf) 2 + 2h(f) + b : 2 (h<f> 2 4- b + g (f>+f) : b<fr + %f(f> + c, 
giving (f> 2 : cf) : 1 proportional to linear functions of 1, 8 + x , $x, and therefore a quadric 
relation (*P%, 8 + x, 1) 2 = 0, with coefficients wdrich are not in general (a, b, c, /, g, h). 
Suppose, however, that the coefficients have these values, or that the correspondence is 
(a, b, c, f, g, K$dx, 8 + x, B 2 = °> 
we must have 
TWO VARIABLES. 
(a, b, c, f, g, hQacfy 2 4- 2h<f> + b, — 2 (/u£ 2 + b + g </> + /), b(f> 2 + 2\f<f> + c) 2 = 0, 
that is, 
(ac + b 2 + 2bg — 4fh) (a, b, c, f, g, h\<f> 2 , — 2(f), l) 2 = 0, 
or, we have 
ac + b 2 + 2bg — 4fh = 0, 
ematics, vol. xn. (1873), 
as the condition in order that the symmetrical (2, 2) correspondence between 8 and x 
may be the same correspondence as that between 8 and cf), or between <f) and x• 
l-and-circu inscribed polygon 
Math. Jour., t. xi. (1871), 
(2, 2) correspondence, and 
(8, x) will have a (not in 
, to a given value 8 there 
nd the values 8, of ^ 
% Xa °I X (viz- one of the 
there correspond the two 
rrespond two values of 8; 
and say 0 2 ; that is, the 
etrical. 
(«-*);