949] 
THEORY OE CURVES IN SPACE. 
471 
cone il=0 of the order 12 in 36 lines each twice, which lines are consequently the 
nodal lines of the cone il = 0. The mere condition of the 36 lines lying upon a 
cone of the order 6 shows that the 36 lines are not arbitrary; and we have moreover, 
through the 36 lines, cones of the orders 7, 8, 9, 10 and 11 respectively. Obviously 
the foregoing reasoning is quite general, and for the surfaces of the orders [i, v we 
have (as stated above) the cone il of the order fxv, with h = \\iv (/x — 1) {v — 1) nodal 
lines, the intersections (each counting twice) of this cone with a cone of the order 
n = (+ ~ 1) ( v — 1) 5 and moreover the h nodal lines lie also in cones of the orders 
n + 1, n + 2, ..., n + /1+ v — 2 respectively. 
To examine the meaning of the theorem, I form the table 
+ 
V 
d 
h 
n, ..., n + ¡i + v — 2 
2, 
2 
4 
2 
1, 2, 3 
2, 
3 
6 
6 
2, 3, 4, 5 
2, 
4 
8 
12 
3, 4, 5, 6, 7 
2, 
5 
10 
20 
4, 5, 6, 7, 8, 9 
3, 
3 
9 
18 
4, 5, 6, 7, 8 
3, 
4 
12 
36 
6, 7, 8, 9, 10, 11 
3, 
5 
15 
60 
8, 9, 10, 11, 12, 13, 14 
4, 
4 
16 
72 
9, 10, 11, 12, 13, 14, 15 
4, 
5 
20 
120 
12, 13, 14, 15, 16, 17, 18, 19 
Here ¡x, v = 2, 2, there are 2 nodal lines, which are arbitrary, and of course lie on 
cones of the orders 1, 2, 3 respectively. So /x, v = 2, 3; there are 6 nodal lines, 
which are not arbitrary, inasmuch as they lie on a cone of the order 2; but regarding 
them as arbitrary lines on such a cone, we can through them draw a cone of the 
order 3 or any higher order, and it is thus no specialisation to say that they lie 
upon cones of the orders 3, 4, and 5. But going a step further ¡x = 2, v = 4: here we 
have 12 nodal lines which, inasmuch as they lie on a cone of the order 3, are not 
arbitrary: and they are not arbitrary lines upon this cone, for they lie on a cone 
of the order 4, and such a cone can be drawn through at most 11 arbitrary lines 
on a cubic cone. In fact, upon a cone of the order 0, taking at pleasure N lines, 
the condition that it may be possible through these to draw a cone of the order 
0 + 1 is (0 + 1) (0 + 4) = N + 3 at least; for if this number were =iV"+ 2, then through 
the JSf points we have only the improper cone (x + fiy + <y Z ) U e = 0, if U e =0 is the 
cone of the order 0. It thus appears that the 12 nodal lines are not arbitrary lines 
on a cubic cone, but that they constitute the complete intersection of a cubic cone and