374] 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
523 
the case in hand, this is 8' + k' = 2 (I obtain this by the consideration of a quartic 
curve, having a cusp of the second kind, and also a double point; 8 + k has here its 
maximum value = 3; and as the double point gives 8 = 1, the cusp of the second 
kind gives 8' + k — 2) ; and joining to the former conditions this new condition, we have 
definitely S' = 1, k = 1, t— 1, ¿' = 1. I have elsewhere noticed, [343], that the cusp of 
the second kind was equivalent to a double point and cusp, and accordingly proposed 
to call it the node-cusp; but I had not then remarked that it was also necessary to 
treat the tangent as equivalent to a double tangent and a stationary tangent (or 
inflexion). 
It appears from the foregoing considerations, that any singularity whatever is to 
be regarded as equivalent, and that in a perfectly definite manner, to a certain number 
8' of double points, k of cusps, r of double tangents, and l of inflexions; we have 
only to ascertain how for any given singularity the values of these numbers are to 
be ascertained; and when this is done, Pliicker’s equations will be applicable to any 
singularities whatever of a plane curve. 
At any point of a plane curve there is either one branch, or any number of 
branches, touching or not touching each other: taking the given point as origin, then 
for each branch the equation of the curve gives for the ordinate y an expression of 
the form 
y — Ax p + Bx q + ..., 
where the series is arranged in ascending powers of x, and where the coefficients 
A, B,... have definite unique values; and, conversely, that which is given by such 
expression of y is a branch of the curve. It is assumed that the axis of y, or line 
x = 0, is not a tangent to the curve; this implies that the exponents p, q,... are none 
of them inferior to 1, or, what is the same thing, that the lowest exponent p is = 1 at 
least: it is for the most part convenient to take the axis of x, or line y = 0, a tangent 
to the branch; the lowest exponent p is then > 1. 
The exponents may be all integer, and the branch is then said to be linear; or 
else the exponents or some of them may be fractional, and the branch is then 
superlinear; viz. in the latter case, assuming that the fractional exponents are all of 
them in their least terms, and that a. is the least common multiple of the denominators 
i 
(so that the expression for y is a rational function of x a ), then the branch is quadric, 
cubic, &c. according as we have a = 2, a = 3, &c. It is clear that the expression for y 
i 
has precisely a values, viz. the values obtained by attributing to the radical x a each 
of its a values. Corresponding to each of these a values, we have what I term a 
partial branch of the curve, so that the quadric branch is made up of two partial 
branches, the cubic branch of three partial branches and so on; for a linear branch 
or when a= 1, a partial branch is nothing else than the branch itself; and the 
expression a partial branch will accordingly include the case of a linear branch. 
Suppose that at any point of the curve we have two partial branches, belonging 
or not belonging to the same branch; let these be referred to the same axes, the 
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