472
ON HALPHENS CHARACTERISTIC n.
[949
a quartic cone. But through such 12 lines we may draw cones of the 5th and higher
orders, and it is thus no further condition that the 12 lines lie on cones of the
orders 5, 6 and 7 respectively.
So again ¿¿ = 3, v = 3; we have here 18 nodal lines which, inasmuch as they lie
on a cone of the order 4, are not arbitrary: and they are not arbitrary lines on
this cone inasmuch as they lie also on a cone of the order 5, and such a cone can
be drawn through at most 17 arbitrary lines on the quartic cone: it thus appears
that the 18 nodal lines are 18 out of the 20 lines of intersection of a quartic
cone and a quintic cone. But there is no further condition, for through such lines
we can draw a cone of the order 6 or any higher order and thus the lines lie
on cones of the orders 6, 7 and 8 respectively. It appears probable however that,
for higher values of /¿, v, it would be necessary to take account not only (as in
these examples) of the cones of the orders n and n+ 1, but of those of higher orders
n + 2, &c.; and thus that it is not the true form of the theorem to say that the
h nodal lines must be h out of the n (n + 1) lines of intersection of two cones of the
orders n and n +1 respectively.
It appears, by what precedes, that the h, — lines which are the
nodal lines of the cone of arbitrary vertex which passes through the curve oi inter
section of two surfaces of the orders /¿, v respectively, form a remarkable special
system of lines, which well deserve further study. I remark also that, without having
proved the negative, it seems to me clear that given the values of d, h, n it is
only in the cases where the h lines form some such special system (and not in the
general case where the h nodal lines are any lines whatever on a cone of the order
n) that there exists a curve (d, h, n); and thus that the question for further
investigation is, for given values of (d, h, n) to determine the conditions necessary for
the existence of a curve in space with these characteristics (d, h, n).
