578 CORRECTIONS AND ADDITIONS TO THE MEMOIR [415
2. In the formulae
q = b 2 — b — 2k — 37 — 6t,
r = c 2 — c — 2h — 3/3,
it is assumed that the nodal curve has no actual multiple points other than the
t triple points, and no stationary points other than the y points which lie on the
cuspidal curve; and similarly that the cuspidal curve has no actual multiple points,
and no stationary points other than the /3 points which lie on the nodal curve; and
this being so, q is the class of the nodal curve and r that of the cuspidal curve,
But we may take the formulae as universally true; viz.. q may be considered as
standing for b 2 — b — 2k — 3y — 6t, and r as standing for c 2 — c — 2h — 3/3; only then q
and r are not in all cases the classes of the two curves respectively.
3. In the formulae No. 6 et seq., introducing the new singularity to, we have as
follows:
(a—b — c)(n — 2) = (k — B — 0 + 2co) — 6/3 — 4y — St,
{a-2b- 3c) (n - 2) (n - 3) = 2 (8 - G - Sco) -8k- 18h - 12 (6c-3/3-2y - i) ;
and substituting these in n' — a (a — 1) — 2b — 3c, and writing for n! its value
= a (a — 1) — 28 — 3k, we have, as in the memoir,
n! — n{n— l) 2 — n(7b + 12c) + 4 6 2 + 8 b + 9c 2 + loc
- 8k - 8h +18/3 + 12y + 12i - 91
-2G- 3B - 30;
viz. there is no term in to.
Writing {n — 2) (n — 3) = a + 2b + 3c + (— 4n + 6) in the equations which contain
(n — 2)(n — 3), these become
a (— 4>n + 6) = 2 (S — C) — a 2 — Ip — 9<r — 2j — 3% — loco,
b (— 4>n + 6) = 4<k — 26 2 — 9/3 — 6y — 3i — 2p —j,
c (— 4>n + 6) = 6h — 3c 2 — 6/3 — ly — 2i — 3cr — x ~ 3co,
(Salmon’s equations (C)); and adding to each equation four times the corresponding
equation with the factor (n — 2), these become
a 2 - 2a = 2 (8 - G) + 4 (« — B) — a - 2j — 3* — 3to,
26 2 -2b =№-¡3 + 6y + 12t-3i + 2p-j,
3c 2 — 2c = 6h + 10/3 + 4 9 — 2 i + 0 a — ^
Writing in the first of these a 2 -2a = n' + 28 +3>c-a, and reducing the other two
by means of the values of q, r, the equations become
n - a = - 2C - IB + k - a - 2j - 3x~ 3to,
2q + /3 + 3i+j —2 p,
3r + c + 2i+ x = 5a + /3 + 4<0 + co.
