These straight lines do not belong to the surfaces, of which the 
equations arise from J/—O by leaving out the first or second 
column; therefore we have found : 
Through every base-point pass 35 tangents which have in that base- 
point a five-point contact. 
So the locus of the groups of (n—4) points S, which every t, 
with point of contact B has moreover in common with the corre- 
sponding surface ”, has a 35-fold point in 4. A plane passing 
through B contains 6(7—4) points Son the generatrices of (t,)° 
lying in it; the locus in question is therefore a twisted curve of 
order (6n4-11). : 
We can now find the number of tangents 4,5, which have with 
a Pr in B a four-point contact and elsewhere a two-point contact. 
To that end we consider the correspondence between the planes, 
which project two points S and 3S’, lying on the same surface d” 
out of an arbitrary axis a. Any plane 6 passing through « contains 
(6n+11) points S, is therefore associated to 624-11) (n—-5) planes 
0’, which each project a point SS’. On a generatrix of the cone (¢,)° 
which is intersected by a, lie (n—4)(n —5) pairs S,S’; the plane 
(at,) replaces therefore (n—4)(n 
(o, 0’). The remaining coincidences arise apparently from coincidences 
S = M; their number amounts to 2(62+411)(n—5)—6(n—4)(n—5)= 
=(6n+46)(n—5). 
There are consequently (67+46)(n—5) tengents tis, which have in 
5) coincidences of the correspondence 
B a four-point contact. 
The method followed here will, for the sake of brevity, be indi- 
cated as “process (1). 
§ 2. The straight lines ¢,, which have in £ three coinciding 
points in common with the surface 
7 
aa, = by -{- Yer = Oy, 
are indicated by the two equations 
n—1 : nl n—l 
ady dr + Bb, by + Yer Cy =0, 
> 
n—2,2 ? 
aay, Ay a Bb, lope Yer ¢ 
1—2 
By elimination of «, 8, y from these equations we find, that the 
points @, in which each surface ” is moreover intersected by its 
two principal tangents ¢,, are lying on a surface of order (n+-3). 
As any é, bears a group of (n—3) peints Q, the surface (Q)+ has 
a sextuple point in 4; the tangents in B apparently form the cone (¢,)°. 
By a plane p containing B, (Q)’+% is intersected in a curve gt, 
We shall now consider the pairs of points (Q, Q’ lying on the straight 
