CURVE. 
475 
785] 
In further illustration of the Pliickerian dual generation of a curve, we may con 
sider the question of the envelope of a variable curve. The notion is very probably 
older, but it is at any rate to be found in Lagrange’s Théorie des fonctions analytiques 
(1798) ; it is there remarked that the equation obtained by the elimination of the 
parameter a from an equation f(x, y, a) = 0 and the derived equation in respect to a 
is a curve, the envelope of the series of curves represented by the equation f (x, y, a) = 0 
in question. To develope the theory, consider the curve corresponding to any particular 
value of the parameter ; this has with the consecutive curve (or curve belonging to 
the consecutive value of the parameter) a certain number of intersections, and of 
common tangents, which may be considered as the tangents at the intersections ; and 
the so-called envelope is the curve which is at the same time generated by the points 
of intersection and enveloped by the common tangents ; we have thus a dual gener 
ation. But the question needs to be further examined. Suppose that in general the 
variable curve is of the order m with 3 nodes and k cusps, and therefore of the class 
n with t double tangents and ¿ inflexions, m, n, 8, k, t, c being connected by the 
Pliickerian equations,—the number of nodes or cusps may be greater for particular values 
of the parameter, but this is a speciality which may be here disregarded. Considering 
the variable curve corresponding to a given value of the parameter, or say simply the 
variable curve, the consecutive curve has then also 8 and k nodes and cusps, con 
secutive to those of the variable curve ; and it is easy to see that among the 
intersections of the two curves we have the nodes each counting twice, and the cusps 
each counting three times; the number of the remaining intersections is = m 2 —25—3/c. 
Similarly among the common tangents of the two curves we have the double tangents 
each counting twice, and the stationary tangents each counting three times, and the 
number of the remaining common tangents is = n 2 — 2t — Sc (= m 2 — 28 — 3/c, inasmuch 
as each of these numbers is as was seen — m + n). At any one of the m? —28 — 3/c 
points the variable curve and the consecutive curve have tangents distinct from yet 
infinitesimally near to each other, and each of these two tangents is also infinitesimally 
near to one of the n 2 — 2t — Sc common tangents of the two curves ; whence, attending 
only to the variable curve, and considering the consecutive curve as coming into actual 
coincidence with it, the n 2 — 2r — Sc common tangents are the tangents to the variable 
curve at the m 2 — 28 — 3/c points respectively, and the envelope is at the same time 
generated by the m 2 — 23 — 3/c points, and enveloped by the v? —2t — Sc tangents ; we 
have thus a dual generation of the envelope, which only differs from Plucker’s dual 
generation, in that in place of a single point and tangent we have the group of 
to 2 — 23 — 3/c points and n 2 — 2t — Sc tangents. 
The parameter which determines the variable curve may be given as a point upon 
a given curve, or say as a parametric point ; that is, to the different positions of the 
parametric point on the given curve correspond the different variable curves, and the 
nature of the envelope will thus depend on that of the given curve ; we have thus 
the envelope as a derivative curve of the given curve. Many well-known derivative 
curves present themselves in this manner ; thus the variable curve may be the normal 
(or line at right angles to the tangent) at any point of the given curve ; the inter 
section of the consecutive normals is the centre of curvature ; and we have the evolute 
60—2