470 
CURVE. 
[785 
It is easy to derive the further forms— 
(7) 
L — К 
= 3 {n — m), 
(8) 
2(т —8) 
= (n — m) (n + m — 9), 
(9) 
\m (m + 3) — 8 — 2к 
= \n (n + 3) — t — 21, 
(10) 
(m — 1) (m — 2) — 8 — 
K — "2 ( П ~ 1) (w — 2 ) — T — 
(11, 12) 
m 2 —28 — Sk 
= n 2 — 2t — Sl, = m + n, 
the whole system being equivalent to three equations only: and it may be added that, 
using a to denote the equal quantities 3m + t and 3n + к, everything may be expressed 
in terms of m, n, cl. We have 
К — CL — 3 n, 
i —a — 3 m, 
2S = m 2 — m + 8 n — 3a, 
2т = n 2 —n+ 8 7ii — 3a. 
It is implied in Plticker’s theorem that, m, n, 8, к, т, t signifying as above in 
regard to any curve, then in regard to the reciprocal curve n, m, t, l, 8, к will have 
the same significations, viz. for the reciprocal curve these letters denote respectively 
the order, class, number of nodes, cusps, double tangents, and inflexions. 
The expression \m (m + 3) — 8 — 2k is that of the number of the disposable con 
stants in a curve of the order m with 8 nodes and к cusps (in fact that there shall 
be a node is 1 condition, a cusp 2 conditions): and the equation (9) thus expresses 
that the curve and its reciprocal contain each of them the same number of disposable 
constants. 
For a curve of the order to, the expression \m (to — 1) — 8 — к is termed the 
“ deficiency ” (as to this more hereafter); the equation (10) expresses therefore that 
the curve and its reciprocal have each of them the same deficiency. 
The relations m 2 — 2S — 3/c = n 2 — 2r — Si, = to + n, present themselves in the theory 
of envelopes, as will appear further on. 
With regard to the demonstration of Pliicker’s equations it is to be remarked 
that we are not able to write down the equation in point-coordinates of a curve of 
the order to, having the given numbers 8 and к of nodes and cusps. We can only 
use the general equation (*$ж, у, z) m — 0, say for shortness u= 0, of a curve of the 
mth order, which equation, so long as the coefficients remain arbitrary, represents a 
curve without nodes or cusps. Seeking then, for this curve, the values n, l, t of the 
class, number of inflexions, and number of double tangents,—first, as regards the class, 
this is equal to the number of tangents which can be drawn to the curve from an 
arbitrary point, or what is the same thing, it is equal to the number of the points 
of contact of these tangents. The points of contact are found as the intersections of 
the curve u — 0 by a curve depending on the position of the arbitrary point, and 
called the “first polar” of this point; the order of the first polar is =m — 1, and