A Treatise On the Geometry of Surfaces
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The preceding form of a quaternary cubic is due to Sylvester*, ■ and we shall now show how it can be established by means of the foregoing results.
Let J. be a node on the Hessian ; then we have already shown that (i) the polar quadric of A consists of a pair of planes inter- secting in a straight line LMN] (ii) the line LMN lies in the Hessian ; (iii) it has three conic nodes upon it, which we shall suppose situated at the points L, M, N. Let a = 0, u = be the equations of LMN; then the polar ...quadric of A must be of the form aa^ — eu^ = 0.
Integrating with respect to a, it follows that the equation of the cubic surface must be of the form C7 = aa^ + b^ + 07^ + dh^ + eu^ + ^8^ (/i7 + vh) + yS (P72 + 2Q7S + M^) + Gy'^h + Hyh^ = . . .(20).
Let us construct a triangle bv drawing three lines LBD, MGD and NGB in the plane a through L, M and N. Then each of * A quaternary cubic can also be expressed as the sum of six cubes. See Dixon, Proc. Lond. Math. Soc. Series 2, vol. vii. p. 389.
THE HESSIAN OF A CUBIC SURFACE 51 these lines has one degree of freedom in the plane a ; also since a may be any plane which passes through LMN, the plane a has also one degree of freedom.
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To quickly assess the difficulty of the text, read a short excerpt:
The preceding form of a quaternary cubic is due to Sylvester*, ■ and we shall now show how it can be established by means of the foregoing results.
Let J. be a node on the Hessian ; then we have already shown that (i) the polar quadric of A consists of a pair of planes inter- secting in a straight line LMN] (ii) the line LMN lies in the Hessian ; (iii) it has three conic nodes upon it, which we shall suppose situated at the points L, M, N. Let a = 0, u = be the equations of LMN; then the polar ...quadric of A must be of the form aa^ — eu^ = 0.
Integrating with respect to a, it follows that the equation of the cubic surface must be of the form C7 = aa^ + b^ + 07^ + dh^ + eu^ + ^8^ (/i7 + vh) + yS (P72 + 2Q7S + M^) + Gy'^h + Hyh^ = . . .(20).
Let us construct a triangle bv drawing three lines LBD, MGD and NGB in the plane a through L, M and N. Then each of * A quaternary cubic can also be expressed as the sum of six cubes. See Dixon, Proc. Lond. Math. Soc. Series 2, vol. vii. p. 389.
THE HESSIAN OF A CUBIC SURFACE 51 these lines has one degree of freedom in the plane a ; also since a may be any plane which passes through LMN, the plane a has also one degree of freedom.
What to read after A Treatise On the Geometry of Surfaces?
You can find similar books in the "Read Also" column, or choose other free books by Alfred Barnard Basset to read onlineMoreLess
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