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; a = 6. 
The equations of the spinode curve are 
W(X + Y + Z)* + XYZ=0, 
X 2 + F 2 + Z--2YZ-2 ZX -2 XY = 0, 
viz. the curve is a complete intersection, 2x3. 
Each of the mere lines is a single tangent (as at once appears by writing for 
instance W = 0, X =0, which gives (Y—Z) 2 = ()); that is, /3' = 3. 
Reciprocal Surface. 
154. The equation is found by means of the binary cubic 
4>(T-xU)(T-yU)(T-zU) + wT 2 U, 
viz. writing for shortness 
/3 = x + y + z, 
7 = yz + zx + xy, 
8 = xyz, 
then the cubic function is 
(12, w- 4/3, 47, -128$T, Uf, 
and the equation of the reciprocal surface is found to be 
432 S 2 
+ 
64 7 3 
- 
(w — 4/3) 3 S 
+ 
72 (w — 4/3) 78 
— 
(w — 4/3) 2 7 2 = 0 ; 
expanding, this is 
w 3 . — 8 
+ 
w n -. 12/38 — 7 2 
+ 
8w . — 6/3 2 8 + /37 2 + 978 
+ 
16 (4/3 3 8 — /3 2 7 2 — I8/378 + 47 3 + 278 2 ) = 0 ; 
or substituting for /3, 7, 8 in the first and last lines, this is 
w 3 . — xyz 
+ 
w 2 . (12/38 — 7 2 ) 
+ 
8iv . — 6/3 2 8 + Sf + 978 
+ 
16 (2/ — z) 2 (z — x) 2 {x — yY — 0 
(where /3, 7, h=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 — 0, z-x = 0; w- 0, x-y- 0 
(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), 
(x = 0, y= 0), reciprocals of the facultative lines; in fact, in regard to {y, z) conjointly 
7 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.