308 
ON THE THEORY OF INVOLUTION. 
[348 
Secondly, when the curves are such that there are two critic centres which 
become ultimately a twofold centre. 
And, thirdly, when the curves are such that there are two critic centres which 
remaining distinct from each other belong ultimately to the same critic curve. 
38. The curves 1 and 2 in the left-hand figures respectively represent the nodal 
curves corresponding to slightly different values of k, which in the right-hand figures 
respectively give the curve corresponding to a twofold value of k. In the first pair 
of figures the curves U= 0 and V=0, about to touch in the left-hand figure, touching 
in the right-hand figure, are shown by dotted lines. It will be observed that in the 
second case in the left-hand figure the two nodes which give rise to a cusp are the 
one of them an acnode and the other a crunode; this is in fact the only mode of 
drawing the figure so that a cusp shall present itself. The transition of form is one 
of ordinary occurrence in cubic curves and in curves of a higher order; thus if 
y* = (x — a) (.x -b)(x- c), where a <b< c, then if a — b, we have an acnodal curve, 
if b = c a crunodal curve, and if a — b = c a cuspidal curve. 
39. In the case of two conics, n = 2. We have here simply □ = R, where R = 0 
is the condition in order that the two conics may touch each other. The nodal curves 
are of course the three pairs of lines passing through the points of intersection of 
the two conics, and the nodes of these curves, or critic centres, are the centres of 
the quadrangle formed by the four points in question; or, what is the same thing, 
they are the system of conjugate points common to the two conics, viz. the points 
which are such that the polar of one of them taken with respect to either of the 
conics is the line joining the other two of them. The diacritics are any conics 
passing through the three points.