By Beniamino Segre
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Additional resources for Some Properties of Differentiable Varieties and Transformations: With Special Reference to the Analytic and Algebraic Cases
Example text
IT! we have thus obtained altogether n - 1 independent projective invariants. The various results enunciated above can easily be proved analytically, after having introduced suitable coordinates in 5 w This, at the same time, affords simple expressions for those invariants and verifies that they are dependent only on differential elements with order:;;; n. We limit ourselves here to indicating that the most convenient non-homogeneous reference system is one in which the origin is 0, the axes X1> x 2 , ••• , x, are independent lines through 0 chosen in the spaces 51> 52' ...
B. SEGRE [93J, pp. 1:. It will suffice to establish this result in the case of two infinitely near V%'s. 1:, lying between the given pair. 1: infinitely near to it, and by Sk+1' 8 k +1 the ambient spaces of Vi, Vi respectively. Since, by hypothesis, V% is non-degenerate, Sk+1 does not touch Q, and so it possesses a definite polar Sn-k-2 which is skew to it (and therefore also skew to 8 k +1> which is infinitely near to Sk+1)' We fix in any manner a point P of V%. 1: which passes through P. 1:, is subordinate to the perspectivity with vertex Sn-k-2 between the two ambient spaces Sk+1> 8 k +1 (skew to Sn-k- 2): and this is sufficient to prove the theorem.
Further we choose in Sn a fixed hyperplane 71:, not passing through 0, and a variable hyperplane X' also Invariants of Contact and of Osculation 40 not passing through 0, which tends to a limit XO through 0, but not containing the line 51. When X is sufficiently close to Xo, the hyperplane X cuts Land L' in distinct points, Q and Q', uniquely determined in the neighbourhood of O. We denote by P and Ri the points in which n and (}i are cut by the line QQ', and we put ()ij= (QQ' R i P)n+r-2i+l/(QQ' R j P)n+,-2i+l, where i =1= j and i,j = r + 1, r + 2, ...