The collected mathematical papers of Arthur Cayley.

432

A MEMOIR ON CUBIC SURFACES.

[412

The complete intersection with the cubic surface is made up of the rays each twice
and of a residual sextic, which is the spinode curve; ' = 6.
The equations of the spinode curve are
W (X + Y + Z)2 + XYZ= O,
X2 + Y+ Z- 2YZ- 2ZX - 2XY = 0,
viz. the curve is a complete intersection, 2 x 3.
Each of the mere lines is a single tangent (as at once appears by writing for
instance W=, X=0, which gives (Y Z)2= 0); that is, /3'=3.
Reciprocal Surface.
154. The equation is found by means of the binary cubic
4 (T- xU)(T-y) (T-zU)+wT2U,
viz. writing for shortness
P=x+y+z,
/3 = x + y + z,
7 =yz + zx + ay,
8 = xyz,
then the cubic function is
(12, w - 4/9, 4y, - 128$T, U)3,
and the equation of the reciprocal surface is found to be
432 82
+ 64 y3
- (w- 4/3)3 8
+ 72 (w-4/3)y8
- (w - 4/)2 7y2 = 0;
expanding, this is
wU3. -
+ w2. 12/38- y2
+ 8w.- 6/28 +/3 72 + 978
+ 16 (4/338 _ -2/ - 18/y8 + 473 + 2782)= 0;
or substituting for /3, y, 8 in the first and last lines, this is
w. - xyz
+ w2.(1288- y2)
+ 8w. - 6/2 +/382 + 9y8
+ 16 (y - z)2 ( - x)2 (x _ y)2 0
(where /3,, = x + y + z, yz + zx + xy, xyz). The section by the plane w =0 (reciprocal
of the unode) is made up of the lines w=0, y-z= 0; w=O, z-x=0; w=0, x-y=O
(reciprocals of the rays) each twice.
155. The nodal curve is at once seen to consist of the lines (y = 0, z = 0), (z = 0, x = 0),
( = 0, y= 0), reciprocals of the facultative lines; in fact, in regard to (y, z) conjointly
y is of the order 1, and 8 is of the order 2; hence every term of the equation is
of the order 2 in y, z; and the like as to the other two lines: b= 3 as above.