[314 
314] ON THE CURVES SITUATE ON A SURFACE OF THE SECOND ORDER. 
71 
HE SECOND 
-38.] 
generating lines, 
m with 
any 
two 
quations 
of 
the 
hird and fourth, 
= 0, and 
if 
the 
ctively are suit- 
‘der (or 
say 
for 
rded as 
the 
CO- 
= 1-^ • 
V 
ym' 
X ’ 
P ’ 
Xp’ 
(X, p, v, 
P) 
are 
ighout borne 
in 
are essentially 
ed with. 
ud homogeneous 
urface; and this 
the curve the 
this equation, expressed in terms of the coordinates (X, p, v, p), is 
AXp + Bpp + Gv\ + Dpv = 0; 
or, as it is more conveniently written, 
( G, D $X, p) (v, p) = 0; 
A, B 
and if from this and the equation of the curve we eliminate X : p or v : p, say the 
second of these quantities, we obtain 
(*$X, p)P(-A\-Bp, G\+Dp)i = 0, 
which is of the order p + q in (X, p); and X : p being known, v : p is linearly deter 
mined. There are thus p + q systems of values of the coordinates, or the plane meets 
the curve in p + q points; that is, the curve is of the order p +q. 
A linear equation A\ + Bp = 0 gives a generating line, say of the first kind, of 
the quadric surface, and a linear equation Gv + Dp = 0 gives a generating line of the 
second kind: and by combining the one or the other of these equations with the 
equation of the curve, it is at once seen that the curve meets each generating line 
of the first kind in q points, and each generating line of the second kind in p points. 
Consider the curves of the order n: the different solutions of the equation p + q = n 
give different species of curves. But the solution (n, 0) gives only a system of n 
generating lines of the first kind, and the solution (0, n) gives only a system of 
generating lines of the second kind. And in general the solutions (p, q) and (q, p) 
give species of curves which are related, the one of them to the generating lines of 
the first and second kinds, in the same way as the other of them to the generating 
lines of the second and first kinds; and they may be considered as correlative members 
of the same species. The number of distinct species is thus 4 (n — 1) or \n, according 
as n is odd or even; for n = 3 we have the single species (2, 1) or (1, 2); for n = 4, 
the two species (1, 3) or (3, 1), and (2, 2); for n= 5, the two species (4, 1) or (1, 4), 
and (3, 2) or (2, 3); and so on. Thus for n = 3, the species (2, 1) is represented by 
an equation of the form 
(a, b, c$X, pf v + (a', b', c'][X, p) 2 p = 0, 
which belongs to a cubic curve in space. To show d posteriori that this is so, 
I observe that the equation expressed in terms of the original coordinates (x, y, z, w) is 
x (a, b, c§x, y) 2 + z (a!, b', cy) 2 = 0, 
which by means of the equation xw — yz— 0 of the quadric surface is reduced to 
(a, b, c\x, y) 2 + axz + 2b'yz + c'yw = 0 ; 
and this is the equation of a quadric surface intersecting the quadric surface 
xw — yz = 0 in the line x — 0, y — 0; and therefore also intersecting it in a cubic curve.