( 720 ) 
each of the three points of contact of a right line d can be formed 
when D' coincides with YD" the number of threefold tangents cutting 
g” on the curve dr is but the third part of the number of the indi- 
cated coincidences of (4), thus equal to 
+n (n—4) (n—5) (n—6) §(n —2) (2n?+-5n+8) — (n—1) (n+2) (n—3)} = 
En (n—A4) (n—5) (n—6) (n?+3n?—2n—12). 
This is at the same time the order of the curve (D) formed by 
the points D which the threefold tangents d have still in common 
with o”. 
Now we can also find the order z of the seroll (d). This scroll 
being touched by ©” in the points of (C) and being cut in the points 
(D) we have namely 
ne = n (n—2) (n —4) (n—S) (n?-++-5n-+12) + 
+n (n—A4) (nN—5) (n—6) (n?+3n?—2n—12). 
Out of this we find 
The threefold tangents of a” form a scroll the order of which is 
zn (n—3) n—A4) (n—5) (n?-+-3n—2) *). 
§ 8. To find the degree of the spinodal curve I consider the pairs 
of principal tangents a, a’ of which the common point of contact A 
lies in the plane «. If two rays s and s of a pencil (S, 6) are con- 
jugate to each other, when they rest on two right lines a and a’, 
then in (S,6) a symmetric correspondence with characteristic number 
n (dn —4) is formed. The coincidences can be brought to three groups. 
First « and a’ can cut the same ray s; their plane of connection 
is then tangential plane, their point of intersection A lies on the 
polar surface of ‚S. Such a ray s coincides with two of the rays s' 
conjugate to it. So the first group contains 7” (# — 1) double coinci- 
dences. 
Secondly s can cut the curve a; then too it coincides with two 
rays s. So the second group consists of 7 double coincidences. 
Finally a single coincidence is formed when « coincides with @’. 
The number of these coincidences evidently amounts to 27 (Bn —4) — 
2n (n — 1) — 2n = dn (n — 2). From this ensues : 
The parabolic points form a twisted curve (spinodal line) of order 
An (n — 2). 
1) In Satmon-I'tepter we find on page 638 by mistake 7?-+-32-+ 2 instead of 
ning. 
On page 643 we find the derivation of the number of fourfold tangents and of 
the numbers of tangents 45, ¢s,9,9 and {5,5 
