943 
A,: a C,, with a twelvefold point in D,, 
an eightfold point in D,, 
uitetold. points «in in 10 ea eed ee 
a sixfold point in J, ; 
Ao,: a Care with a n(n+1)-fold point in D,, 
a (n’—1)-fold point in D,, 
u -tolde points in Ds Dian 
an n(n—1)-fold point in: B; 
D 
8? 
Aa: a C3243,41 With an n*-fold point in D,, 
an (nv? + n—1)-fold point in D,, 
Ge 4-°n)-fold points. an: Ds WO ee 
an (n + 1)-fold point in B, 
8° 
We prove this as follows. It goes without saying that the locus 
of A, is the line D,4,. Through any point 4, of this line on ecurve 
u, of (3’) passes and this curve is cut by 4,D, for the third time 
in A,. In D, we draw the tangent to uw, and we indicate by A,’ 
the point common to this tangent and D,4,. Now if u, describes 
the pencil (3’) it will happen twice that A, and A,’ coincide; in 
each of these two cases A, coincides with D,, so that D, is a 
double point of the locus of A,. This point A, describes a rational 
cubic curve, to be indicated henceforth by a,, any line through D, 
having only one more point in common with this curve. It contains 
Hie points 19), Di. Dias DB, cuts each of the lines Die 
Pee Dan one point. 
Let us now consider the locus of A,. It is immediately evident 
that D, is an ordinary and JD, a threefold point of this locus; for 
a, is cut by Bb, D, in only one, by 4, D, in three points; in the 
same manner we prove D,, D,...., D, to be double points. So 
we have still to investigate how many times A, coincides with B, 
Let A, be once more an arbitrary point of «, and wu, the curve of 
(3’) through A,; then the tangent of v, in B, cuts a, in three points 
A. So the points A, and A’, generate a correspondence (1,3) 
furnishing — @, being rational — 4 coincidences. Any coincidence 
of A, and A,’ gives a coincidence of A, and B,; so A, describes 
a curve of order seven, to be indicated henceforth by @,, any line 
through B, containing three points more of this locus. 
We can prove that B, is a fourfold point of @, also as follows. 
In ease A, coincides with one of the points A,', A, is at the same 
time the tangential point of B, on the curve out of (3’) through A, 
So the number of points common to @, and the tangential curve of 
B, — i.e. the locus of the tangential point of 5, on any curve 
