136 
NOTE ON THE NODAL CURVE OF THE 
[333 
33 
which curve is in fact an excubo-quartic,—viz. a quartic curve the partial intersection 
of a quadric surface and a cubic surface, having in common two non-intersecting right 
lines. To show that this is so, I remark that the coefficients a, b, c, d, e, qua linear 
functions of the four coordinates, satisfy a linear equation which may be taken to be 
a-\-b~\-c~\~d-\-e — 0^ 
this being so, the first form shows that the curve in question lies on the quadric 
surface 
ac — b 2 + ^ (ad — bc) + -$ (ae + 2bd — 3c 2 ) + \ (be — cd) + ce — d 2 = 0, 
or, as this equation may also be written, 
c(a — ^b — % c — £ d + e) — b 2 ad + £ (ae + 2bd) + \be — d? = 0. 
Substituting for c its value, this equation is 
— (a + e + b + d) (f a + § e) — b 2 + £ ad + £ (ae + 2bd) + ^ be — d 2 = 0, 
or, what is the same thing, 
9 (a 4- e + b -(- d) (a + e) + 6 (b 2 + d 2 ) — 3 (ad + be) — (ae -f- 2bd) = 0. 
Hence, finally, the equation of the quadric surface is 
9a 2 + l7ae + 9c 2 + 66 2 — 2bd + 6d 2 + 9ab + 9de + 6ad + 6be — 0 ; 
and the curve lies also on the cubic surface 
ad 2 — b 2 e = 0. 
It only remains to show that these surfaces have in common two right lines, and 
to find the equations of these lines. 
The cubic surface is a skew surface or “ scroll ” such that the equations of any 
generating line are d — 6b = 0, e — 6 2 a = 0, where 6 is an arbitrary parameter. But 
considering the two lines 
(d — 9 1 b — 0, e — 6{~ a = 0), (d — 6 2 b = 0, e — 6 2 2 a = 0), 
the general equation of the quadric surface through these two lines may be written 
A . (d-6 1 b)(d-6,b) 
+ B . (e — 6 2 a) (e — 6 2 2 a) 
+ C . (d — 0x b) (e — 6. 2 2 a) + (d — 6Jb) (e — 6da) 
+ a ^ n [(d — 6ib) (e — 6 2 a) — (d — 6 2 b) (e — df a)} = 0 
or, 
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