944 
@te de) amounts to four, the common points coinciding with the 
base points of (37) disregarded ; for, the tangential curve is of order 
four and admits B, as threefold point whilst it passes only once through 
D,, D,,...D,. So B, is a fourfold point of a, and this curve is of 
order seven. From the number of the double points we deduce that 
a, is rational; this is right, for it corresponds point by point with 
the line D,B,. 
As to the locus of A, it is immediately ciear that B, is a double 
point and D, a threefold point on it, while it passes four times 
throuch: DDA DE 
The tangential curve of D, is cut by «, besides in the base points 
in 6 points more, which implies that D, is a sixfold point on the 
locus of A, and that this curve is of order twelve, any line through 
D, containing six more points of it. In the same manner we deter- 
mine the loci of the points A,, A,, A,, etc. and then the loci of 
As, and Ao,+1 can be found by the Bernoullian method. Provision- 
ally we only still wish to remark, that the locus of A, has an 
eightfold point in J,, for this proves that the points B, and D, 
form two Steinerian points of order three on 8 curves of (8). 
§ 9. Let us return to the point we started from. We have seen 
that the curve j, under discussion — the locus of the ninth threefold 
point — is at the same time the locus of the points each of which 
forms with B, a Steinerian pair of the third order. On each curve 
of (8) lie besides the base points eight points more of j,; moreover 
D,, D,,..., D, are eightfold points of jz. 
We have now to investigate whether B, lies on j, or not. This 
can only happen if on a curve xv, of (B) the point B, coincides with 
one of the eight points each of which forms with it a Steinerian 
pair of the third order. However it is easy to prove that a suchlike 
coincidence of two Steinerian points can only present itself in a 
node; for the group of the nine inflexions this is immediately evident 
and for tbe other groups of Steinerian points of the third order it 
can be deduced from this by projection. Now B, is not a node of 
a curve out of (2); so it does not lie on jz. 
As the number of points common to jr: and w, amounts to 72 
we find: 
“The curve jr is of order twenty-four; it has D,, D,,...,D, as 
eightfold points.” 
$ 10. We will enumerate some points of 7,, which curve will 
be denoted furthermore by j,,. It is cut by the line D,D, in eight 
