374] 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
527 
equation P 1 P. 2 ...P a = 0: the last-mentioned equation breaks up into the equations 
P t = 0, P. 2 = 0,... P a — 0 ; and selecting for example the equation P 2 = 0, this gives the 
system P 1 = 0, P. 2 P 3 ... P a ^P 1 = 0, or since we require only the intersections at the 
singular point, and AP X = 0 does not pass through this point, this may be replaced 
by Pj = 0, P. 2 P 3 ... P a = 0. The complete system is thus (P x = 0, P 2 P 3 ... P a = 0), 
(P2 = 0, P 1 P 3 ...P a = 0),...(P a = 0, P 2 P- 2 ... P a _i = 0); or, what is the same thing, we have 
each pair (P r — 0, P s = 0) taken twice. To eliminate y from these equations, we have 
merely to write P r — P g = 0, or, what is the same thing, we have £(P x , P 2 ... P a ) = 0, 
£ denoting the product of the squares of the differences of the functions (P 1} P 2 ...P a ). 
Suppose that any two partial branches P r = 0, P s = 0 intersect (according to the 
above-mentioned definition) in p points; then P r — P s contains the factor xP, and hence 
the product £(P ly P 2 ...P a ) contains as a factor x to the power 2Sp, that is, the 
equation in x has 2Xp roots each = 0. Whence if 2p = \M, then the equation in x has 
M roots each = 0, or the curve and polar have at the singular point M intersections, 
that is M = 28' + 3k . 
I have no complete proof to offer of the remaining equation k = a — 1, it was 
obtained from the consideration of a particular case as follows. Consider the linear 
branch y = AxP + ... , where the exponents are all positive integers, and taking the 
axis of x to be the tangent, the least exponent p is greater than unity; if p = 2 
there is at the origin no inflexion, if p — 3 there is a single inflexion, and generally 
the number of inflexions is =p> — 2. Now it will presently appear that in line-coordi- 
p 
nates the equation of the branch is Z — A'X p ~ l , or replacing Z, X by the original 
p 
point-coordinates y, x the branch y = A'x p ~ l + ... has at the origin p — 2 cusps; but 
in the branch in question we have a=p — 1, and the number of cusps is thus 
= a — 1 ; this result is confirmed by other particular instances, and I assume in 
general that we have k = a — 1; whence in the case of a simple singularity, or where 
there is only one branch we have M = 2<f + 3k, k = cl — 1, or, what is the same thing, 
8' =\ \_M — 3 (a — 1)], k=ol — 1. The reasoning is easily adapted to the case of a com 
pound singularity. 
I consider the branch 
y + Ax p + Bx q + ... = 0, 
(where it is assumed that the axis of x is a tangent to the branch, and therefore 
that the lowest exponent p is greater than unity), introducing the coordinate z for 
homogeneity, this becomes 
yz~ Y + Ax p z~ p + Bx^z~i + ... = 0, 
and I proceed to find the corresponding equation in line coordinates, taking these to 
be X, Y, Z, we have 
XX =pAx p ~ l z~ p + Bqx q ~ l z~ q + ..., 
\Y= z~\ 
\Z = — yz~“ 2 — pAx p z~ p_1 — qBx q z~ q ~ 1 + ...,