The collected mathematical papers of Arthur Cayley.

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INVESTIGATIONS IN CONNEXION WITH CASEY'S EQUATION.

71

and substituting for P, Q, R, dU,, d, dW their values, the left-hand side is =-Jdz,
which is = 0; hence the equations in question are proved, and (f, g, h) is a point
on the curve A.
It is to be noticed, that the two curves X, A are geometrically connected through
the three arbitrary points as follows: viz. taking as axes the sides of the triangle
formed by these three points, then starting from any point (f, g, h) of A, we take the
inverse point (, -, 1 ), the harmonic line thereof fx + gy + hz =0, and finally the
inverse conic -- + + = 0, which by what precedes touches X in the point corresponding to the assumed point (f, g, h) of A: and conversely starting with an assumed
point on X, we take the conic +f + -h = 0 which passes through the angles of the
triangle and touches E at the assumed point; the inverse line fx + gy + hz= O; the
harmonic point (h,, -1, ) of this line; and finally the inverse point (f g, h), which
will be on the curve A, the point corresponding to the assumed point on the curve E.