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Extension of a Geometrical Porism and other Theorems

Published online by Cambridge University Press:  24 October 2008

J. H. Grace
J. H. Grace
Affiliation:
Peterhouse
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Extract

It is a familiar fact in Solid Geometry that the problem of finding a twisted cubic curve with four given tangents is poristic. In fuller statement four prescribed tangents are apparently sufficient to determine such a curve yet

(A) In general there is no twisted cubic touching four given lines,

(B) If there is one there is an infinite number.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 4 , October 1929 , pp. 421 - 432
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* I have discussed the connexion between A and B in § 5.

* Of course the result is quickly obtained by the methods of enumerative geometry. I have deliberately preferred the above close imitation of the well-known method of finding the common intersectors of four lines in three dimensions.

* The poristic condition is two-fold for three invariants reduce to one and there are ∞2 solutions if there is one. This agrees with experience, but, as far as I know, there is no logical support for the general principle involved.

* The equation of the curve may be written

which is the poristic condition already mentioned (§2). It may be remarked that there is a mutual moment I 12 of two planes in five dimensions and with the same notation we are led to the poristic condition

satisfied when four planes osculate a C 5: but the condition being two-fold this is incomplete.

* By a geometrical problem is meant a problem of discovering a geometrical entity subject to conditions which are prima facie just sufficient to determine it.

* It might be said once for all that the number is zero in the first: my reason for objecting to this will be found in § 7.

† Cf. Enriques, , Lezioni sulla teoria geometrica delle equazioni, 2 (1918), 317–21, and the references there given.Google Scholar

‡ Though the profane might say that C is B stated in a form more suitable for the introduction of tacit assumptions: the merit (and defect) of most geometrical reasoning.

* A 51 is where [h 2h 3h 4] meets h 5 and as this [5] cuts the curve in a 2, a 3, a 4 each twice its equation is easily written down.

* Proc. London Math. Soc. (2), XXVIII (1928), p. 328.Google Scholar

† I.e. the sextie giving the feet of the osculating primes of C 6 through the point.

* That in a [7] the ∞1 [3]'s which meet five lines do so in projective ranges follows from the fact that between ten points (here two on each line) in the [7] there are two linearly independent syzygies.

* These results indicate the unique determination of the parameters from the figure.