( 653 ) 
as the harmonie conjugate of re, =O with respect to the two tangents 
of the double point 5. 
Indeed, the first polar curve of the point A with respect to &, 
breaks up: into 7, =O and the conic 
mos + nav, — 0, 
which has in the vertex A five points in common with /,. Evidently 
the only simple tangent of 4,, passing through the cusp A, is repre- 
sented by 
bs = ie zel) RT Nn tE WES 
2 
The equation of a conic, which touches &, on z,=0 in the 
cusp A and still in two other points, can be written in the well 
known form : 
zw," + 2e(me,* + nz,x,) + 2’, (a’«, — be) — 0. 
If such a conic degenerates into two straight lines, one of which 
will be the tangent 2, — 0, then the other must be the only double 
tangent belonging to 4, 
If we put 2me = —1, it follows from the last equation 
DANE B Oe ee «ar (A) 
and we have the equation of the double tangent d. 
From the form of this equation is evident, that the double tangent 
passes through the point of intersection of the lines 
d==Oand az, — br, — 0: 
We can now say: The line (BL) joining the double point (2) 
to the point (/?) of intersection of the double tangent with the simple 
tangent (@, passing through the cusp (A), is the harmonic conjugate 
of the line AS with respect to the two tangents of the double point B. 
2. A pencil of conics having the two common tangents x, — 0, 
2, = 0, with the points of contact A and BS respectively, is indicated 
by the equation 
fee EE ee eoa 2) 
Each of these conics cuts the curve /, moreover in two points 
M, N; let us determine the equation of the right line MN. 
By eliminating 2°, out of the equations (k,) and (1), we find 
z,—0, «*,=0, and MN = (n—um) «, + u@a,—be,)—0 (2) 
so all these lines MN, passing through the point R, determine a 
pencil [A]. We put 
men . . . . . ° . . 
(n—um)y © 6) 
