516 
ON A SPECIAL SEXTIC DEVELOPABLE. 
[373 
In my theory of the singularities of curves and torses, Liouville, t. x. (1845) 
pp. 245—250, [30], translated under the title “ On Curves of Double Curvature and Deve 
lopable Surfaces,” Cambridge and Dublin Mathematical Journal, t. v. (1850), pp. 18—22, 
[83], I omitted to take account of a noteworthy singularity, viz. this is, the stationary 
tangent line; or when the system has three consecutive points in a line, or, what is 
the same thing, three consecutive planes through a line. I reproduce the theory with 
this addition as follows. We have 
to, 
r, 
n, 
a, 
ß 
% 
9 
h 
x 
y 
the order of the system, = order of the curve, 
„ rank of the system, = class of curve, = order of torse, 
„ class of the system, = class of torse. 
„ number of stationary planes, 
„ „ stationary points, 
„ „ stationary lines, 
„ „ lines in two planes, 
„ „ lines through two points, 
„ „ points in two lines, 
„ „ planes through two lines. 
This being so, the 
section of the torse by an arbitrary plane is a plane curve for which 
r 
n 
x 
m -{- ^ 
9 
a. 
is the order, 
„ class, 
„ number of nodes, 
„ „ cusps, 
„ „ double tangents, 
„ „ inflexions; 
and we have thence Plticker’s six equations, which may be considered as included in the 
three equations 
n = r (r — 1) — 2x— 3 (m + ^), 
a = Sr(r — 2) — 6x — 8 (to + '&-), 
r = n (n — 1) — 2g — 3a. 
Similarly considering the cone standing on the curve and having an arbitrary point 
for vertex, then for this cone 
is the order, 
„ class, 
„ number of nodal lines, 
„ „ cuspidal lines, 
TO 
r 
h 
ß 
y 
n -1- '5- 
double tangent planes, 
inflexions ;