NOTES AND REFERENCES. 
615 
nter- 
the 
er of 
have 
1er d 
gives 
ough 
moid 
e is 
of the orders n +1, n respectively, such that the equation of the monoid surface is 
w =jp. Or what is the same thing, we have always a monoid surface of the order 
n +1: we thus arrive at the notion of Halphen’s characteristic n. 
Instead of the foregoing equation kd = 2Ji + 6, we thus have nd = 2li + 6, and for 
given values of d, h there is thus a minimum value of n (viz. nd must be at least 
= 2h); there is also a maximum value of n, viz. this is the least value for which 
\ n (n + 3) is = or < h, for with such a value of n there is always through the h 
nodal lines a cone of the order n. 
For a given value of d, we have h = at most ^ (d — 1) (d — 2), and Halphen 
shows that h must be at least = [4 (d — l) 2 ], if we denote in this manner the integral 
part of the expression within the brackets. And then, h having any value between 
these limits, for any given values of d, h we have by what precedes a certain number 
of values of 11. 
We thus have primd facie curves in space of the several forms (d, h, n): but it 
may very well be, and in fact Halphen finds that when d is = 9 or upwards, then 
for certain values of h, 11 as above, there is not any curve (d, h, n): thus d = 9, h= 17, 
the values of 11 are 11 = 4 or 5, but there is not any curve d = 9, h = 17, for either 
of these values of n; or say the curves (9, 17, 4) and (9, 17, 5) are non-existent. 
And I notice further that in certain cases for which Halphen finds a curve 
(d, h, 11) such curve does not exist except for special configurations of the h nodal 
lines not determined by the mere definition of n as the order of the cone of lowest 
order which passes through the h nodal lines: for instance d — 9, A = 16, 11= 4 for 
which Halphen gives a curve, I find that it is not enough that the 16 nodal lines 
are situate on a quartic cone, but that they must be the 16 lines of intersection of 
two quartic cones. 
I remark moreover that Halphen does not carry out the foregoing principle of 
classification according to the values of (d, h, 11): thus d = 9, h = 22, the values of 11 
are 6 and 5; viz. the 22 nodal lines are in general on a sextic cone but they may 
be on a quintic cone; the curves (9, 22, 6) and (9, 22, 5) exist each of them, but 
he gives only the former of the two forms. The form (9, 22, 6) has a capacity 36 
(depends upon 36 constants) but (9, 22, 5) a capacity 35 only, and I assume that 
Halphen considered it as a particular case of (9, 22, 6), (there is it seems to me a 
want of precision in his definition of a famity)—but I consider that this is an 
abandonment of the principle—the two curves differ ipso facto in that in the first 
form the 22 nodal lines are not, in the second form they are, on a quintic curve. 
I11 Nother’s theory the characteristic n does not present itself. 
Resuming the general theory, and considering d, li, 11 as given, we start from the 
cone U = 0 of the order d, with h nodal lines lying in a cone of the order n: we 
take P = 0 a cone of the order 11 passing through the k nodal lines, and besides 
meeting the cone U= 0 in 6 lines; nd = 2h + 6, (where 6 may be = 0). And we then 
have Q = 0 a cone of the order 11 +1 passing through the h lines and the 6 lines;