( 632 ) 
besides the base points of (3’) three more points in common, which 
points to the fact that three times one and the same point is at the 
same time tangential point of B, and D,. If, however, B, and D, 
have on a curve out of (3’) the same tangential point, then D, will 
lie on the curve j,. This last will be cut by «, in each of the 
points D,, D,...D, three times and once in three other points; 
so it is of order nine. 
If D,, D,...D, are points chosen arbitrarily, then the locus of 
the point which can be the ninth double point of a curve of order 
six already possessing double points in D,, D,...D, is a curve jn of 
order nine with triple points in D,, D,...D.- 
Moreover we have found the following generation of ), : 
If we determine on a curve u, out of the c,-pencil (B’) with the 
base points D,, D,..,D, the points having the same tangential points, 
as the ninth base point B,, then if u, describes the pencil (8’) these 
points will describe the curve jn. 
8 
§ 7. We shall now show analytically that the curve j, is of 
order nine and possesses triple points in D,, D,,...D,. 
To this end we regard the net of curves yp = w, + hu? , +uv’, = 0, 
where w,— 0 represents a curve of order six with double points in 
D,, D,...D,, whilst u,=O and v, =O are the equations of two 
cubic curves through those eight points. The curves of the net 
passing through an arbitrary point form a pencil; we choose the pencil 
of curves passing through an arbitrary point J of j,. In $5 we 
have seen that in this pencil appears one curve possessing in Ja 
ninth double point; therefore : 
Each point of j„ is the ninth double point of one of the curves 
contained in the net (»). 
§ 8. We take an arbitrary triangle V, 0,0, as triangle of coordinates ; 
the locus of the double points of the net v= w, + 4u*®, + + uv’, 
= 0 is then found by elimination of 2 and u out of the equations 
a Ae eS and ES 
dz, dz. de, 
As equation of that locus we then find : 
dw (du dv du dv du (du dv du dv . 
jk ea dz, dz, de,de,) da,\da,dz, dz, dz, 
dw {du dv du, 
Be en 
de, \de, de, dw,de = 
The factor uv in the first member of this equation means simply 
