314 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
special case is however in the present memoir treated irrespectively of the general one, 
and the equation for the critic values of k is found to be of the order 3; this of 
course means that the equation of the order 12 breaks up into two equations of the 
orders 9 and 3 respectively, but I have not attempted to show how the decomposition 
and reduction arise. Moreover, in the general case the equation, Disc 4 . Disc 4 . (U+kV)—0, 
which is the condition for the existence of a twofold critic value, presents itself in 
the form RQ*P 2 — 0, where R = 0 is the condition that the two cubics (U = 0, V — 0) 
shall touch each other; Q = 0 the condition that there shall be in the involution 
U+kV=0 a curve having (not a mere node but) a cusp; and P = 0 the condition 
that there shall be a curve having two nodes, or (what is the same thing) breaking 
up into a line and conic. But in the special case, which, as already noticed, is here 
considered irrespectively of the general one, the equation Disc 4 . Disc 4 . (U + kV) — 0, for 
the existence of a twofold critic value presents itself in the reduced form Q = 0, giving 
the condition, that corresponding to the twofold critic value there shall be a curve 
having (not a mere node but) a cusp. 
Article Nos. 1 to 18, Explanations, Definitions, and Residts. 
1. I consider the involution 
xyz + k(x+y + zf (fix + py + vz) = 0, 
where ¿c = 0, y= 0, z = 0, x + y + z = 0 may be considered as representing any four lines 
no three of which meet in a point, and Xx + py + vz = 0, as representing any fifth line 
whatever: k is a variable parameter. The lines x + y + z = 0, Xx + py + vz = 0, are a pri 
mary line and a satellite line of any cubic of the series, viz. the tangents x = 0, y = 0, 2 = 0, 
at the points of intersection with the primary line x + y + z = 0, meet the cubic in 
three points lying on the satellite line Xx 4- py + vz = 0. 
2. A critic value of & is a value for which the corresponding curve 
xyz + k (x + y + zf (fix + py + vz) — 0 
has a node; and such node, or say rather the site of such node, is a critic centre. 
3. The critic values of k are in effect determined by a cubic equation, and the 
coordinates of the critic centre are then given rationally in terms of k\ there are 
consequently three critic values of k\ and the same number of critic centres, and of 
nodal curves: it is however found to be convenient to express as well the critic value 
of k, as the coordinates of the critic centre, rationally in terms of an auxiliary para 
meter 6 which is given by a cubic equation. 
4. The cubic equation in k (or what is the same thing, that in 6) may have a 
twofold root (pair of equal roots); or, say rather, it may have a twofold root and a 
one-with-twofold root: corresponding to the twofold value of k we have a twofold critic