476 
ON THE THEORY OE THE EVOLUTE. 
[363 
the point (X, Y, Z); the number of such points is the number of normals which can 
be drawn through a given point (X, Y, Z), viz. it is equal to the class of the evolute. 
The points in question are given as the intersections of the two curves U = 0, 0 = 0, 
which are respectively curves of the order in, hence the number of intersections is 
= m 2 . It is to be observed, however, that if the curve U= 0 has nodes or cusps, then 
the curve 0 = 0 passes through each node of the curve U = 0, and through each cusp, 
the two curves having at the cusp a common tangent; that is, each node reckons for 
two intersections, and each cusp for three intersections. Hence, if the curve U= 0 
has 8 nodes and k cusps, the number of the remaining points of intersection is 
= m 2 — 28 — 3/c. The class of the evolute is thus = m 2 — 2S — 3k. The number of 
inflexions is in general = 0. If, however, the given curve touches the absolute, then it 
has been seen in a particular case that the effect is to diminish the class by 1, and 
to give rise to an inflexion, the stationary tangent being in fact the common tangent 
of the curve and the absolute: I assume that this is the case generally. Suppose 
that there are 0 contacts, then there will be a diminution = 0 in the class, or this 
I will be = m 2 —28— 3/c—0; and there will be 0 inflexions; there may however be 
special circumstances giving rise to fresh inflexions, and I will therefore assume that 
the number of inflexions is = i. 
■ 
Suppose in general that for any curve we have 
to, the order, 
n, „ class, 
8, „ number of nodes, 
k, „ ,, cusps, 
t, „ „ double tangents, 
l, „ „ inflexions. 
Then Pliicker’s equations give 
l — k = 3(ii — to), r — 8 = \{n — m) (n + to — 9) ; 
and we thence have 
l - k + r — 8 = £ (w — 1) (n — 2) — \ (m — 1) {m — 2), 
nr, what is the same thing, 
\ (m — 1) (to — 2) — 8 — k = — 1) (n — 2) — t — i. 
Now M. Clebsch in his recent paper “Ueber die Singularitäten algebraischer Curven,” 
Crelle, vol. lxiv. (1864), pp. 98—100, has remarked (as a consequence of the investi 
gations of Riemann in the Integral Calculus) that whenever from a given curve another 
curve is derived in such manner that to each point (or tangent) of the given curve 
there corresponds a single tangent (or point) of the derived curve, then the expression 
\ (m — 1) (m — 2) — 8 — k, = ^ (n — 1) (n — 2) — t — i,