ON THE HIGHER SINGULARITIES OF A PLANE CURVE.
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[374
and the tangent at this point to g double tangents; hence, if there is no other point
singularity, the equations give
n= to (to — l) — 2g,
t — 3m (to — 2) — Qg,
8 + g = \m (m — 2) (to 2 — 9) — (to 2 — m — 6) 2g + 2g (g — 1),
the last of which may also be written
8 = \m (m — 2) (m 2 — 9) — (to 2 — to — g — 4^) 2g.
And Nos. 77—82, pp. 217—222. For a cusp of the second kind, we have
n = m(m — 1) — 5,
l = 3m (to — 2) — 15,
8 = \m (to — 2) (to 2 — 9) — (to 2 — to — 7) 5;
these equations Plticker establishes by an independent algebraical investigation, and
having done so, he remarks that they are deducible from the foregoing ones by writing
therein g= 2£; that is, that the cusp of the second kind may be considered as
equivalent to 2| double points, and the tangent at the cusp to 2\ double tangents.
And he thence passes to the cusp of a higher cusp equivalent to li + £ double points
and h +1 double tangents. The results in this general case (although not, as in the
original case, g = 2-|, established independently) is perfectly correct; but the theory is
open to a grave objection.
I remark, that assuming a certain singularity to be equivalent to the numbers 8'
of double points, k' of cusps, t of double tangents, and t of inflexions, we have in
the first instance to determine 8', k, r and l in such manner as to give in the
class n, and in the numbers t of inflexions and t of double tangents, the reductions
actually given by the singularity in question. Thus in the case of the cusp of the
second kind, we ought to have
28' + 3k' = 5,
68' + 8/e' +1' = 15,
(to 2 — to — 6) (28' + 3k) — 28' (8' — 1) — 68V — §k' (k — 1) + t' = (to 2 — to — 7) 5,
or, what is the same thing,
28' (8' — 1) + 68V + £k (k — 1) — r = 5 ;
and so in general there are, for the determination of the four quantities 8', k, t, l,
three equations. In the particular case these are satisfied by the values 8' = 2£, k — 0,
t = 2^, i = 0, which are Pliicker’s values; they are also satisfied by the values
8' = 1, k = 1, t =1, t = 1, which have the advantage of being integer instead of
fractional.
But there is really a further condition to be satisfied, viz. the number 8' + k
must have a certain definite value dependent on the nature of the singularity; for