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On Models of Cubic Surfaces by William Henry Blythe

On Models of Cubic Surfaces by William Henry Blythe

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William Henry Blythe
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On Models of Cubic Surfaces

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Book review

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1905 edition. Excerpt: ...cubics have degenerated into straight lines, their degree is the same, and we obtain the same number of intersections. Hence we may infer that there are two common secants and not more than two. Reye finds the twenty-seven straight lines on a cubic surface by correspondence. Let a cubic surface be generated by three projective pencils at centres S, S', S", and let any plane through S correspond to a straight line in a plane P. We see that this is possible, for any plane in the pencil S may be fixed by its intersection with P in a straight line. Now every plane of the pencil S corresponds to planes in the other pencils which by their intersection fix a point on the surface. Therefore we infer that unless the three corresponding planes intersect in a straight line every point on the surface corresponds uniquely to a straight line in P. Next take another plane P' so that every point in P' corresponds to a straight line in P. This is reciprocal correspondence, for as two points in P' lie on a straight line, so the corresponding straight lines in P intersect in a point. We may take as an example the properties of pole and polar. We finally arrive at the conclusion that to every point on the surface corresponds one, and only one point on P', provided the three corresponding planes of the pencils meet at a point. I. Every straight line on P' corresponds to a twisted cubic on the surface, for every straight line on P' determines a number of straight lines through a point on P, which in turn determine three axial pencils, which by their intersections fix a twisted cubic on the surface. This is not the same kind as the twisted cubic on p. 35, and is said to be of the second species. II. Every plane section of the surface corresponds to a plane...

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