616 
NOTES AND REFERENCES. 
and this being so we have 
consequently U = 0 and w — 
w = -p for the equation of the monoid surface, and 
for the equations of the curve, viz. the cone U= 0 
and the monoid surface of the order n +1 meet in the h lines each twice, in the в 
lines, and in the curve of the order d\ (n + 1) d= 2h + в + d. Observe here that the 
cone Q = 0 as a cone of the order n + \ subjected only to the conditions of passing 
through the h lines and the в lines has in general a capacity = \ (n + 1) (n + 4) — h — в ; 
this number should be = 3 at least, for if it were = 2, we should have Q = (x + fi у + yz) P 
(since P = 0 is a cone of the next inferior order through the same h + в lines), and 
thus the curve would be a plane curve. Observe further that the cone U= 0, qua 
cone of the order d with h nodal lines has in general a capacity = d (d + 3) — h ; the 
cone P = 0, by what precedes may be regarded as determinate, and the cone Q = 0 as 
just appearing has in general a capacity = \ (n +1) (n + 4) — h - в ; there is a term 
+1 for the implicit constant factor in the function Q, and we thus find for the 
capacity of the curve the expression \d (d + 3) — h +1 + %{n + 1) (n +4) — h — 0, viz. this 
is = I d (d + 3) + (n 2 + on) + 3 — nd, = \(d — n? + ^ (3d + 5n) + 3, which putting for a 
moment d—n = a is = ^ a 2 + ^ (8Й — 5a) + 3, = 4<d + ^ (a — 2) (a — 3) ; hence restoring for 
a its value, we find capacity of curve = Ы + %{d — 2 — n) (d — 3 — n) : in particular if 
n = d — 2 or d — 3, the capacity is = 4P 
We are thus able in the case where £ (n + 1) (n + 4) - h — 6 = 3 or more, say 
^ n (n + 5) = or > h + 6 + 1, actually to construct the equation of a curve (d, h, ri), 
having in the case where n = d- 2 or d- 3 a capacity = 4d: the conditions in 
question for any given value of d, are satisfied by the considerable number of curves 
which form Halphen’s “premier groupe.” 
For instance d = 9, then the complete table of the values of h, n, 6 is 
d h n в Cap. 
16 
4 
4 
38 
5 
13 
0 
17 
4 
2 
0 
5 
11 
0 
18 
4 
0 
36 
5 
9 
36 
19 
5 
7 
36 
20 
5 
5 
36 
21 
5 
3 
36 
6 
12 
0 
22 
5 
1 
35 
6 
10 
36f 
23 
6 
8 
36f 
24 
6 
6 
36f 
25 
6 
4 
36t 
26 
6 
2 
36f 
27 
6 
0 
36f 
28 
7 
7 
36f