AJ/JB : AI/IB = A′J′/J′B′ : A′I′/I′B′.
But, by § 17,
AI/IB = A′J′/J′B′ = −1;
therefore the last equation changes into
AJ · A′I′ = BJ · B′I′,
that is to say—
Theorem.—The product of the distances of any two corresponding points in two projective rows from the points which correspond to the points at infinity in the other is constant, viz. AJ · A′I′ = k. Steiner has called this number k the Power of the correspondence.
[The relation AJ · A′I′ = k shows that if J, I′ be given then the point A′ corresponding to a specified point A is readily found; hence A, A′ generate homographic ranges of which I and J′ correspond to the points at infinity on the ranges. If we take any two origins O, O′, on the ranges and reduce the expression AJ · A′I′ = k to its algebraic equivalent, we derive an equation of the form αxx′ + βx + γx′ + δ = 0. Conversely, if a relation of this nature holds, then points corresponding to solutions in x, x′ form homographic ranges.]
§ 39. Similar Rows.—If the points at infinity in two projective rows correspond so that I′ and J are at infinity, this result loses its meaning. But if A, B, C be any three points in one, A′, B′, C′ the corresponding ones on the other row, we have
(AB, CI) = (A′B′, C′I′),