( 659 ) 
therefore a value invariable for all the curves k, with a rhamphoid cusp 
and a node, if the same value of # for all such curves has been chosen. 
We see yet also, that the four points of intersection of the curve 
k, with each line A, passing through #, this point R, and the common 
point to kh and x, =O are three pairs of the same involution. Then 
the pair 7, =O, x, =O is a degenerated conic of the pencil of conics 
(1) which bears /, with the projective pencil [£]. 
7. A line passing through the point (M) of intersection of the 
double tangent (d) and the cuspidal tangent (rz, = 0) has an equation 
of the form 
Pin taney ee Ee OP Og) yg ey a, (LL) 
If we eliminate d out of (14) and (£,}, writing the equation of 
k, in the form 
ML. —NL.L,)? Tr (A4mnz,—a’a ON ho | i (7 
3 epe 23 1 2 3 4 
we shall obtain 
wee,” — (ma’?,—nw,x,)? = CG 
therefore 
uws, E (me? nvt) = 0 EE dl) 
To any ray of tbe pencil || corresponds a pair of conics (15), 
which form an involutory system for all values of u. The two 
conics of the conjugate pair have in the vertex B a pair of tangents 
PE ENE, 
which is divided harmonically by the two lines 7, =0, 2, == 0. All 
the conics of the involutory system osculate one another in the 
Busp-Á on z, = 0. 
From this follows an other generation of £,. 
Let be gwen an involutory pencil of conics, which osculate 
each other in a point (A) on the common tangent (7, = 0), 
and a pencil of rays [M | having its verter (M) on the 
tangent x,=0, then we can arrange a correspondence 
between these pencils in this manner, that the parameter of 
a ray im the pencil [M | is the square of the parameter 
belonging to the corresponding conjugate pair of conics in 
the involutory pencil. 
The two pencils generate a curve of order five, which breaks 
up into the common corresponding right line «,=0O and the curve 
k, of the considered species. 
If we choose under all these conjugate pairs of conics that, for 
