The collected mathematical papers of Arthur Cayley.

573] NOTE ON THE (2, 2) CORRESPONDENCE OF TWO VARIABLES. 95
or, what is the same thing, we have: -(o+ ) = (a,..., 1, 0)2:2(a,...,1, 1, 0o0, 4, 1): (a,...0o, s, 1)2
= a02+2h++b: 2(h 2+b+go+f): b2+2/f+c,
giving 02: b: 1 proportional to linear functions of 1, 0 +, OX, and therefore a quadric
relation (*^0a, 0 + x, 1)2 = 0, with coefficients which are not in general (a, b, c, f, g, h).
Suppose, however, that the coefficients have these values, or that the correspondence is
(a, b, c, f, g, h, X, 0 + X, 1)2 =,
we must have
(a, b, c,f, g, h3aSb2+2h +b, -2(h2 +b+g++f), b 2+2f + c)2 = 0,
that is,
(ac + b2+bg - 4fh) (a, b, c,; g, h02, - 2, 1)2= 0,
or, we have
ac + b2 + 2bg - 4fh = 0,
as the condition in order that the symmetrical (2, 2) correspondence between 0 and X
may be the same correspondence as that between 0 and 6, or between Sb and X.