946 
§ 11. We now treat in a summary way the general case: 
o is an arbitearypsnumber 
Once more the arbitrarily chosen points D,, D,,..., D, are given 
and the question is to determine the locus of the point forming with 
these given points a set of nine g-fold points of a non degenerated 
curve of order 89. In the same way as we have used the results 
obtained for 9@=2 in the solution of the problem for o = 3, we 
can solve the successive cases g = 4,5,... by using every time the 
results obtained in the immediately preceding case. So we consider 
for o = 4 at first a variable c,, with fourfold points in D,,D,,...,D, 
and touching a curve wu, of pencil (3) in a point A; then we 
determine the third point of intersection of u, with the line connecting 
the last two points of intersection of c,, and w,, which point is 
independent of the choice of c¢,,, ete. 
But before we state our results more in detail we wish to make © 
a remark. We find, that any point D, which can present itself as 
ninth o-fold point of a non degenerated cz, must coincide with one 
of the points forming with B, a Steinersan point of order g. The 
locus of the latter peints is a curve ¢3;,-1) with (g*—1)-fold points in 
DD... D,. Now however it is evident that this curve degene- 
rates in several cases. So, if e.g. we consider the case 9 = 6, we 
shall find among the points forming with 4, on a curve of (8) 
Steinerian pairs of order six also the points which form with B, 
Steinerian pairs of order two and of order three. So the curve 
Cy2-1), here of order 105, must break up into j,. j,, and a curve 
of order 72 passing 24 times through D,, D,,..., D,. Now the latter 
curve forms tue locus proper of the ninth sixfold point of a non 
degenerated curve c,,. So the two curves of which the first is the 
locus of the ninth g-fold point, the second that of the point forming 
with B, a Steinerian pair of order e, coincide completely if @ is a 
prime number; if g is no prime number the first curve is a part 
of the second. So we have found: 
“The locus of the ninth o-fold point coincides completely or par- 
tially with that of the points forming with B, on the curves of (B) 
Steinerian pairs of order 9. The litter curve cuts any curve of @) 
besides in the base points in (9? —1) more points, has the pots 
D,, D,,..., D, for (9? — 1)-fold points and is therefore of order 
3(9?—1). The former coincides completely with this curve, if @ is 
a prime; in the opposite case its order and the multiplicity of the 
base points on it can be easily deduced from the corresponding 
numbers of the second curve.” 
