526 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
[374 
tangent of the curve is equivalent to the r double tangents and i inflexions; each 
point of contact is equivalent to ^ [M — 32 (a — 1)] double points and 2 (a — 1) cusps, 
the numbers M, a referring of course to the point of contact in question. 
There is no difficulty in passing to the case of the compound singularity when 
the formulae for the simple singularity, one point, one tangent, one branch, are once 
obtained, and I now go back to the consideration of this case. 
The class of a curve is equal to the number of tangents which can be drawn 
through an arbitrary point: the points of contact of these tangents are given as the 
intersections of the curve with a certain curve, the polar of the arbitrary point in 
regard to the curve; this polar passes through each double point and cusp, the double 
point counting as two points of intersection, and the cusp as three points of inter 
section (this is in fact the theory by which is found the reduction =28 + 3«: in the 
class of the curve). Hence, if the curve has a singularity (8', k, t, if), which to fix 
the ideas may be assumed to be a simple singularity, ‘ one point, one tangent, one 
branch ’ ; then the polar passes through the singular point, the number of intersections 
being 28' + 3«', or if the actual number of intersections be M, then we have M = 28' + 3k. 
It is to be shown that the number M is equal to twice the number of common points 
which the curve has with itself at the singular point, so that the last-mentioned number 
is = ^M. Suppose in the first instance that there is only a single branch, and let the 
branch be given by the equation 
P = y 4- Ax v + Bx q + ... = 0, 
or introducing for homogeneity the third coordinate z, let this equation be 
P = yz~ l 4- Ax p z~ p + Bx q z~ q ... =0, 
and let P 1 = 0, P 2 = 0, ... P a = 0, be the corresponding equations for the component partial 
branches; it is allowable to write P 1 P 2 ...P a = 0 for the equation of the curved). 
Hence if (a, /3, y) be the coordinates of the arbitrary point, or putting in the first 
instance y=l, if (a, /3, 1) be the coordinates, then writing A = aS x + /3S y + S z , the 
equation of the polar is AP X P 2 ... P a — 0, or, what is the same thing, 
P 2 P 3 ... P a AP x + P X P S ... P a AP’ 2 + &c. =0, 
and we have 
AP = a {])AxP~ 1 z~ p + qBafl~ 1 z~ q ...) + (3z~ x — (pAx v z~ p ~ l 4- qBx q z~ q ~ 1 ...), 
or putting z = 1, this is 
A P = a(pAx p ~ x +qBx q ~ x ...) + /3 — (pAx p + qBx q ...), 
and we have thence the values of AP x , AP 2 ...AP a ; the thing to be observed is, 
that the equation A P = 0 is not satisfied (and therefore also each of the equations 
APj = 0, ... AP tt = 0 is not satisfied) by the coordinates x—0, y = 0 of the singular 
point. We have now with the equation AP 1 P 2 ...P a = 0 of the polar to combine the 
1 Of course this is not the equation in its rational and integral form, and on this account the reasoning 
of the text is not free from difficulty; the same remark applies to a subsequent equation.