943 
(S)p of these points, we associate each ray d to the 2(n—8) curves 
ce, to whieh it belongs, and consider the figure produced by the 
projective systems [c*} and (d) thus determined. 
Through a point P passes a pencil of ec"; the base-point B is point 
of contact of (n—3)(n+4) bitangents ; this number is the index of 
[er]. The order of the figure produced now amounts to (72 — 3)(n--4) + 
+ 2n(n—3) = (n—3)(8n-+4). To this the curve (Ld) apparently belongs 
twice; for the order of (S)g we find therefore (n—3)3n-4)—4n or 
3(2+- 1 )(n—-4). 
As every d, outside B, contains 2 (n—3)(n—4) points S, (S)g will 
have in B a multiple point of order 3(n-++-1)(n—4)—2(n—3)(n—4) 
or (n-+9)(n—A4). 
14. Let us now consider the correspondence (MR, MS), if R 
and S lie on the same ray d through B. To each ray MAR belong 
2n(n—4) rays MS, each ray MS determines 3(n-+1)(n—4) rays MR. 
The ray MB eontains 2(n—3) points A, consequently represents 
2n—3)(n—4) coincidences. The remaining ones, to the number of 
(n—4)(2n+3n+3—2n-+6), pass through points R=. So there are 
3(n—-4)(n-+3) rays d, which each touch a cr in B and osculate it 
in a point J; the curve (Rog has consequently a 3(n—4)(n+3)-fold 
point in B, 
Now we pay attention to the symmetrical correspondence of the 
rays, which connect JZ with two points S belonging to the same c”. 
The characteristic number is here 3(n—-+1)(n—4)(n—5), while MB 
represents 2(n—3)(n— 4)(n—5) coincidences. The remaining (n—4)(n —9) 
[6(n+1)—2(n-—3)] lie in pairs on a triple tangent, which bas one 
of its points of contact in B. From this we conclude that the curve 
(R)z22 possesses in B a Un-3) (n—4) (n—5)-fold point. 
15. Let D be node of an c”, ¢ one of the tangents in D, S one 
of the intersections of ¢ with ct. In order to find the locus of S, 
we associate to each nodal c* its two tangents ¢ and determine the 
order of the figure produced by it. The tangents ¢ envelop the curve 
of ZrUTHEN; they form consequently a system with index 38(7—1)’; 
for a pencil contains 3(7—1)? nodal curves. By means of the corre- 
spondence of the series of points, which the two systems determine 
on a line, we now find again the order of the figure produced. 
Considering that the locus of J) belongs six times to it, we obtain 
as order of the curve (S) 3n(n—1)(2n—3) + 6(n—1)?—18(n—1) = 
= 3(n-—1)(2n?—n—8). For n=83 we find 42 for it; the 21 straight 
lines of the degenerate curves must indeed be counted twice. 
