( 238 ) 
which is connected, as will be proved, with the theory of double 
points of curves of higher order. 
Let A® again be a conic of the pencil (1 23 4); we imagine 
two of the points of intersection of A? with K® to be constructed, 
take A, and A,, and moreover the pole A,. of A, Ag in reference 
to K? (see diagram). 
The tangents A, Ajo and A, Aj. intersect the conic A? for the 
second time in the points P, and P,; in the same way we can also 
determine the tangents in the points A; and A, with their second 
points of intersection P,; and P4. If A® describes the whole pencil, 
P,, Po, Ps, Ps generate a locus; at the same time the poles 
Aj, ... Az4 generate the locus C° found formerly. The conics 
forming the solution of the problem proposed sub (3) must now be 
brought through the points of intersection of the curve C3 with 
the locus of the points P,, Ps, Ps, Ps. 
10. To determine the order of the locus lastly named, let us 
take a straight line / and determine how many points it has in 
common with it. We take a point A, on J, draw from that point 
two tangents to K? and construct the conic (1234) through 
each of the points of contact; as we can construct two conics, four 
points of intersection A’, A's, A's, A's on J will be found. So to 
one point A, belong 4 points A’ in the just found correspondence. 
Reversely if we construct the conic passing through A‘), then it 
also passes through one of the other points A’ say A',; it determines 
4 points of intersection with A*; the tangents to A? through those 
points determine still three points A,, As, A4 besides A,. Conse- 
quently 4 points A’ correspond to one point A and 4 points A to 
one point A’. 
So there exists a projective correspondence (4,4) between these 
points A and A’ which possesses 8 points of coincidence. So the 
required locus intersects / in 8 points, hence it is a curve of the 
8 order. 
11. However, this curve breaks up into two parts. It is clear 
that AK? itself belongs to the locus of the points of intersection of 
the tangents to A? with the variable conic 42. The remaining curve 
will be of the 6 order; we have now to investigate its particular 
points. These are the following: 
a. The points 1, 2, 3, 4 are double points of K6. To prove this 
we consider point 1, from which we draw the tangents ¢, and t to 
K?. There is a conic of the pencil (1234) passing through the 
