Suppose P=u:v:w and W=x:y:z are points in homogeneous coordinates (e.g., barycentric or trilinear) in the plane of a triangle. Define
P*Q = vwxx-yzuu : wuyy-zxvv : uvzz-xyww.
If P is fixed and Q variable, then P* is an involution. Someone mentioned that this is a HIRST TRANSFORMATION. Unfortunately the mentioner gave no details and his name is not known to me. As for Hirst, this is Thomas Archer Hirst (1830-1892), described in Historia Mathematica 1 (1974) 181-184.
Hirst's involution yields a kind of conjugate. That is, for any particular point Q, we have P*(P*Q) = Q, in the same vein as isogonal conjugate, isotomic conjugate, and Ceva P-conjugate.
Can someone cite Hirst's introduction of this transformation? Is there a better (short, please!) name for the point P*Q than this: "Hirst P-conjugate of Q"?