The collected mathematical papers of Arthur Cayley.

-404

A MEMOIR ON CUBIC SURFACES.

[412

87. And the lines are
a b c f g equations may be written
0 0 0 0 1 '; (3) X=O, Z=0
1 0 1 0 0 0 (4) X+Z=0, W=O
O O O 0 b 1 (1) X O, Y- Z /b =
O O O -6b 1 (2) X- o, Y+ ZVb 0
O O 0 1 / O (1') Z =0, -X^/a+ Y
o o o 1 -a 0 (2'), X Va + Y-=
I I 1
-/b2 2 a( /a- - 2/- ) (11') but for the other lines the
coordinate expressions are
1 1 1 the more convenient.
/7b ~^/b i aC 2 (-V -Vb) -2 (12')
bi \/b - -,/ 2 2( Va + V ) -2 (21'), 1 1 1
1x __: -7 2 2 (-Va + Vb) -2 (22')
6 ab: V

88. The four mere lines and the transversal are each facultative; the edge is
also facultative, counting twice; p' = b'= 7, t' = 3.
That the edge is as stated a facultative line counting twice, I discovered, and
accept, ca posteriori, from the circumstance that on the reciprocal surface the reciprocal
of the edge is (as will be shown) a tacnodal line, that is, a double line with
coincident tangent planes, counting twice as a nodal line. Reverting to the cubic
surface, I notice that the section by an arbitrary plane through the edge consists of
the edge and of a conic touching the edge at the biplanar point; by what precedes
it appears that the arbitrary plane is to be considered, and that twice, as a nodecouple plane of the surface: I do not attempt to further explain this.
89. Hessian surface. The equation is
(X + Z) XZW + (X - Z)Y2 + (X + Z)(3 a, - - b, 3b7X, Z) = 0.
Combining with the equation
XZW+ (X + Z) (Y2 - aX2 - bZ) = 0,
and observing that from the two equations we deduce
- XZY2 + (X + Z) (aX3 + bZ3) = 0,
it appears that the complete intersection of the Hessian and the surface is made up
of the line X=O, Z=0 (the edge) twice (that is, the two surfaces touch along the
edge), and of a curve of the tenth order, which is the spinode curve; c'= 10.