(712) 
If a linear system (c”);, of ook curves c* is given, we can consider 
the locus of the points P41, where a curve of that system has a 
(k-+ 1)-pointed contact with a right line, passing through the fixed 
point O. 
To determine the order g(#) of the locus (P+!) I consider the 
curves (c"), having in the points P of the right line 7 a k-pointed 
contact with the corresponding right line OP. Each ray OP cuts 
the curve individualized by P moreover in (n — 4) points Q. Each point 
of intersection of / with the locus of the points Q being evidently 
a point P41, the locus (Q) is a curve of order ¢(f). 
The curves of (c”); passing through O form a system (c”),-1. The 
order of the locus of the points P‚ where a c” of this latter system 
has a k-pointed contact with OP is evidently indicated by g(& — 1). 
So on / lie g(k—1) points P for which one of the corresponding 
points Q coincides with QO; in other words the locus (Q) passes 
g(k —1) times through O, so it is of order g(& — 1) + (n — 2). 
To determine ¢(/) we have now the recurrent relation 
g(k) = elk — 1) + (n— 2A). 
From this we deduce 
g(k) =p (1) + 4 (A — 1) (Qn -- k — 2). 
Here g(1) represents the order of the tangential curve, thus (22 — 1). 
So we find 
pk) = 4 (& + 1) Qn —k). 
The locus of the points where a curve c*‚ belonging to a k-fold 
infinite linear system has a (k + 1)-pointed contact with a right line 
passing through a fired point O is a curve of order 3 (k + 1) (2n — B, 
on which O is a &k(k-+1)-fold point. 
For (c")¢ determines on a right line 7 through O an involution of 
order n and rank &. The number of (4 + 1)-fold elements of this in- 
volution amounts to (&—+1)(n — £); that is at the same time the 
number of points Pp, lying on r. Consequently O is an $4(k + 4)- 
fold point on (Pi). 
§ 2. Each ray 7 through a fixed point O is touched by 2 (2 —1) 
curves c” of a pencil (c”); the points of contact 7’ are the double points 
of the involution determined by (c") on +. The curves c” indicated by 
these points 7’ intersect 7 moreover in 2(n—1)(n— 2) points S. 
When 7 rotates round O the points |S will describe a curve which 
I shall call the satellite curve of O. 
This curve passes (n-+-41)(n—- 2) times through O; for if r - 
coincides with one of the tangents out of Q to c” passing through 
