By John Milnor

By way of John Milnor, from the Annals of arithmetic reviews in Princeton college Press. Includes--Elementary proof approximately genuine or advanced algebraic units, The curve choice lemma, The fibration theorem, and six extra chapters.

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**Example text**

On the other hand, the four points of intersection, other than A and A*, of the asymptotic tangents of the surfaces S, S* at the points A, A* lie on two lines through the point P, namely, (10) x2 ± (lf^) "22^33/ x4 = 0, x3 = 0. 719 PROJECTIVE INVARIANTS From equations (9), (10) it follows at once that the cross-ratio of the four lines t, t*, (10), (9) is equal to (11) ±(±1)*(^) 1/4 ) = ±(±l)i71/4. (o, », J^f, \ \t22^33' \ ''11^22/ / Hence we obtain the following projectively geometrical characterization of the invariant I: Let S, S* be two surfaces in ordinary space having a common tangent plane at two ordinary points A, A*; t, t* the harmonic conjugate lines of AA* respectively with respect to the asymptotic tangents of the surfaces S, S* at the points A, A*; P the point of intersection of the tangents t, t*; and s any one of the two lines through the point P on which lie the four points of intersection, other than A and A*, of the asymptotic tangents of the surfaces S, S* at the points A, A*.

K2 and Kf may be respectively written as (7) and (40) The equations of af2Xi + atxxx3 — afx2x3 + k*x\ = 0, where k* is an arbitrary constant. From equations (7) and (40) it follows immediately that the equation of the four lines joining Ol to the four intersections of K2 and Kf is (41) a\x\ + 2a2x\x3 + • • • = 0 , the unwritten terms being of at least order 2 in x3 . It is easily seen that the polar line of any point on Oi0 2 with respect to the four lines denned by equation (41) is the line (8).

Let the polar spaces of the line 00* with respect to the asymptotic hypercones of the hypersurfaces F»_i, V*-\ a t the points 0, 0* be respectively denoted by /„_2, /n*-2, which determine a space tn-3 of n — 3 dimensions in the common tangent hyperplane x B + 1 = 0 . If the n — 2 vertices, other than 0 and 0*, of the system of reference in the hyperplane *n+i = 0 be chosen in the space tn-s, then the invariant / may be reduced to i n n / > » i l \ (n-2)/J (9) where Lnn, Mu are the minors of lnn, mu in the determinants L, M respectively.