( 633 ) 
that each point of the curves 7, and v, counting double can be 
regarded as a double point; as locus of the point ./ we find: 
dw (du dv dv du | dw (du dr du dv 
de, \dz, de, de, da, ; de, dx, de, de, de, a3 
| dw (du dv du dv 
- - is =, |= 8; 
de, \de, de, da, dz, 
This equation of order nine represents the curve j, which for the 
future we shall call 5, 
We now allow the vertex O, of the triangle of coordinates to 
ut 
coincide with D,; the equations of the curves w,, w,, and v, are 
ranged according to the descending powers of z,, written thus : 
ee 2 ii nt Ben, 
Ww, == t*, (ax*, + Abe, 2, Hert) J...=0 
. 2 je pe 
u, = wv’, (ae, + ba,)+...=0 
SH (alle Le me a 
v,==27,(a¢,+ 62, +...=0 
The first member of the equation representing the curve j, evidently 
possesses now no term in which «x, appears to a lower power 
fia, /ihe- sixth > so. J. >is a triple point, of 1, and .D,,D,.....D; 
likewise. 
§ 9. To the curves of order six possessing in D,, D,...D, double 
points belongs one degenerated into the line D,D, and a curve of 
order five having in D,, D,... D, double points and passing moreover 
through D, and D,. The latter is cut by D,D, in three points more 
which must lie on j,; thus on each of the lines connecting D; Dy 
three points of 7, can be indicated. 
Let us suppose a conic D,, D,...D, and a quartic possessing in 
_D,, D, and D, double points and passing also through D,, D,.,. D,; 
then these form together alsoac, with double points in D,, D,...D,; 
the remaining three points of intersection of the two curves lie on 
jy. Thus on each of the 56 conies D; DD, DnDn three points of j, 
are determined. 
Each c, through D,, D,...D, cuts 7, besides in these points in 
three points more; we have already seen how these points can be 
determined. We have also seen that B,, the ninth base point of the 
c, pencil, does not lie on j,; by allowing the vertex O, of the 
triangle of coordinates to coincide with B, we can easily deduce 
this out of the equation of 7, 
§ 10. Let w, be a curve of order six possessing in D,, D,... D, 
double points, whilst u, is the cubic through those points; then by 
