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.
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
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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