( 631 ) 
and 2 points P and Q chosen arbitrarily; the two other base points 
of the pencil P' and Q' are then by this completely determined. 
From the preceding is evident, how we can determine those two 
last points and also, that one of these points e.g. P' is independent 
of Q and the other Q' of P. 
We now again understand by w, an arbitrary cubie through 
D,, D,...D,, whilst also J, J' and J" have the same meaning as 
in §4. We then further regard the c, pencil (8), having D,, D,...D, 
as double points and moreover J and an arbitrary point Q as single 
base points. An arbitrary curve out of (8) will touch wv, in J whilst 
the last base point of (3) lies on the c, through D,, D,...D, and Q 
and is to be determined in the way indicated above; the line 
touching in J the w, as well as an arbitrary curve out of (8) we 
shall call j. If we then draw through J an arbitrary line / and if 
A is a point moving along /, then always through A passes one 
curve a, out of the pencil (3); if we allow A to coincide with J 
then the lines j and / will both have in / two points in common 
with a,. From this ensues that / is now a double point of a, and 
lies therfore on the curve which we have indicated by ),. 
If inversely it is given that / is a point of jn and if we bring a 
c, through D,, D,...D, and J, then J must possess the same 
tangential point on c, as B,. We have then proved: 
If we generate a c, pencil with double points in the points D,, 
D,...D, chosen arbitrarily and single base points in a point J of 
the curve j, and in a point Q chosen arbitrarily, then the curves 
of this pencil have in J a common tangent. In this pencil is included 
a curve, having in J a ninth double point. 
§ 6. We have seen, that on an arbitrary curve u, out of the c, 
pencil having D,, D,...D, and B, as base points lie three points 
of j, ; these points have on wu, the same tangential point 7’ as B, 
We now regard first the locus of 7 when wu, describes the c‚ pencil 
which we shall now call (3). Each line / through B, determines one 
curve out of (3), touching it; so / intersects the indicated locus besides 
in B, in one point. Farthermore this locus has in B, a triple point, 
three curves out of (3') possessing in B, a point of inflexion. The 
point 7’ describes therefore a quartic curve ¢, possessing in B, a triple 
point; the points D,, D,... D. lie also on ¢,, as each of the lines 
B,D is touched by one curve. 
Let 7” be the tangential point of D, on w,, then if again wu, de- 
scribes the pencil (8’), 7’ describes a quartic curve f’; ¢, and ¢’, have 
41* 
4? 
