148 
PROFESSOR CAYLEY’S SECOND MEMOIR ON THE 
P be at a node the curve 0 passes through the node and has therefore at this point two 
intersections with the given curve ; but the second intersection belongs to the other 
branch, and the node is not a united point ; in order to make it so, it is necessary that 
the curve 0 should at the node touch the branch to which the point P is considered to 
belong. The thing appears very clearly in the case of a unicursal curve ; we have here 
two values 0=X, 6=t! answering to the node according as it is considered as belonging 
to one or the other branch of the curve ; and in the equation of correspondence 
(4, l) a (9', l) a = 0, writing 0=X, we have an equation (X, 1)“(0', l) a '=0 satisfied by 0'=\' 
but not by 0 '=\, and the equation (0, l) a (0, l) a ’=0 is thus not satisfied by the value 6 —\. 
The conclusion is that a node qua node is not a united point. 
99. But it is otherwise as regards a cusp. When the point P is at a cusp, the curve 
0 (which has in general with the given curve k intersections at P) has here more than 
k intersections, and (as in this case there is no distinction of branch) the cusp reckons as 
a united point. In the case of a unicursal curve, there is at the cusp a single value 0=X 
of the parameter, and the equation (0, 1)“(0, 1)“'— 0 is satisfied by the value 6 =\. But 
for the very reason that the cusp qua cusp reckons as a united point, the cusp is a united 
point only in an improper or special sense, and it is to be rejected from the number of 
true united points. We may include the cusps, along with any other special solutions 
which may present themselves, under a head “ Supplement,” and instead of writing as 
above a — a — a' = 2£D, write a — a — a'+Supp. = 2£D. 
Before going further I apply the theorem to some examples in which the curve 0 is 
a system of lines. 
100. Investigation of the class of a curve of the order m with § nodes and z cusps. 
Take as corresponding points on the given curve two points such that the line joining 
them passes through a fixed point O ; the united points will be the points of contact of 
the tangents through O ; that is, the number of the united points will be equal to the 
class of the curve. The curve 0 is here the line OP which has with the given curve a 
single intersection at P ; that is, we have k= 1. The points P' corresponding to a given 
position of P are the remaining m — 1 intersections of OP with the curve, that is, we 
have oi=n % — 1 ; and in like manner a=in — 1. Each of the cusps is (specially) a united 
point, and counts once, whence the Supplement is =z. Hence, writing n for the class, 
rve have 11 + 2 ,( 711 — \)-\-z=2D, or writing for 2D its value =m 2 — 3m+2 — 2*, we 
have n=irv 2 —m—2}> — 3x, which is right. 
101. Investigation of the number of inflexions. Taking the point P' to be a tangen- 
tial of P (that is, an intersection of the curve by the tangent at P), the united points are 
the inflexions ; and the number of the united points is equal to the number of the in- 
flexions. The curve 0 is the tangent at P having with the given curve two intersections 
at this point ; that is, k=2 ; P' is any one of the m — 2 tangentials of P, that is, a! =m— 2 ; 
and P is the point of contact of any one of the n — 2 tangents from P' to the curve, that 
is, a =n— 2. Each cusp is (specially) a united point, and counts once, whence the Supple- 
ment is =». Hence, writing 1 for the number of inflexions, we have 
/— (to— 2) — (n— 2)+^=4D ; 
