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If we take two quadratic reguli, a line / meets two generators of each. To / we make correspond the other transversal of the four generators. This involutory transformation of the lines of space is one of three, quite similar in principle.! This case admits a very simple and effective algebraic treatment without the use of hyperspace. We may take for the equations of two non-singular quadrics with real rulings X\ +x2 2 — xi —x£ = 0 and ax? +&x2 2 —cxi — dx£ = 0 . On the former lies the regulus R\ defined by X\ — Xs = m(x4: — x2), Xi+^3 = l/(m(^4+x2)). The Plücker coordinates of a line of this regulus are

The transformation of lines of space by means of two quadratic reguli