941 
§ 6. It is now immediately clear that there are four points in 
which any c, with the threefold points D,, D,,...., D, and passing 
through Y can touch w,, i.e. in any of the four points admitting 
FE as tangential point; likewise that any c, with the threefold points 
oe oe rouching «will cut this ‘curve: im AS 
We now will try to determine X in such a way that it coincides 
with one of the four points of which /’ is the tangential point; in 
that case’ any c, with threefold points in D,, D,,....,D, and 
touching w, in X, will have in X a third point in common with u. 
Let us suppose that the point \ has been determined so as to 
satisfy the condition mentioned; then we can describe in w, closed 
hexagons the successive sides of which pass alternately through 2, 
and Y. If we choose B, as first vertex and /? is the third point of 
intersection of 6,X and w,, then there is a closed hexagon with the 
successive sides 5,B,7, TXY, YB,E, EXX, XB,P, PXB, (fig. 1). 
Fig. 1. 
So the points Y to be determined are the eight points each of which 
forms with BL, on uw, a Steinerian pair of order three. *) 
§ 7. We now choose one point out of these 8 and call it X,. 
If we then require that C, has threefold points in D,, D,..., D, and 
touches uw, in X, we can assume arbitrarily four more points 
kK, L. M. N of this curve, which as we have seen above has in X, 
still a third point in common with wv Provisionally we suppose 
L, M.N to be fixed points but A’ to deseribe a right line 4 through 
1) That B, and the point X satisfying the imposed condition form on ez a 
Steinerian pair of order three can also easily be shown by representing the points 
of uz by means of an elliptic parameter. If B is the parameter value for By and 
x that for the point X taken provisionally at random, we find for the values cor- 
responding to 7, Y and E respectively —23, 23—v and #—8,. So the condition 
that E be the tangential point of X is 33 = 3x. Chiefly for the cases p= 4, 5,.. 
presenting themselves later on the use of this parameter proves to be very convenient, 
Compare CiescH: Vorlesungen über Geometrie (p. 619). 
