ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
522 
[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