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Extension of a Geometrical Porism and other Theorems

Published online by Cambridge University Press:  24 October 2008

J. H. Grace
Affiliation:
Peterhouse

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
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.