1276 
9 (n+1)(n—4)(n—5), formed by the straight lines #23, on which 
neither point of contact in B is lying, the twice to be counted 
cone of order 3 (n-+-3) (n—4) with B, and the thrice to be counted 
cone of order (x-+-9) (n—4) with B, 
The tangents tsss form a complex of order 2n(n— 3)(n—A4)(n—5). 
Here the cone with vertex B is composed of the twice to be counted 
cone of order 2(n+3)(n— 4)(n—5) on which B is one of the points 
of contact, and the cone of order 2(n-+-1)(n—4)(n—5)(n—6), for 
which this is not the case. 
§ 9. We can also determine the order and the class of the 
congruence, formed by the tangents with five-point contact ¢,. 
Any point P is point of contact of eleven four-point tangents. 
For the surfaces ©” passing through P form a pencil of which the 
base-curve passes through P. Let the pencil be represented by 
n n 
aas + Bb, —= 0, 
the point P by (y;), then we find, analogously with § 1, that the 
straight lines ¢, with the point of contact P are indicated by 
n—l ‘ n—2 2 n—3 3 | 
Cy d> Cy Us ay dz | 
, == 
n— 1 n—2 ,2 n—3 ,3 
bn ee Me 
They are obtained as intersections of a cubic cone with a monoid 
of order four, which have the tangent at base-curve in common; 
there are consequently 11 straight lines ¢, with point of contact P. 
On the cone of order 6n (n— 3), which the complex {t,} associates 
to P, the points of contact lie consequently on a curve of order 
(672°—18n+11). 
Each generatrix contains moreover (n—4) points S. The locus of 
S has in P a multiple point, the order of which is equal to the 
number of straight lines ¢, passing through P at surfaces #” of the 
pencil determined by P. An arbitrary point lies on (4n?—6n-+-4) 
(n—3) straight lines ¢, of that pencil’). As the 11 straight lines ¢,, - 
which touch in P, are each to be counted four times, P bears 
(4n?—2n+14) (n—A4) straight lines ¢,, which have their point of con- 
tact outside P. 
The above mentioned curve (S) is therefore of order (42?—2n-++14( 
‘n—4)+6n(n—8)(n-—4) or (10n?—20n-+-14)(n—A4). 
1) See my paper “On pencils of algebraic surfaces”. (These Proceedings VIII, 
p- 29). The class of the congruence [t4] has been given wrongly there; the exact 
number is found in another communication (These Proceedings VIII, p. 817). The 
same observation holds good with regard to the class of the congruence [¢3,9]. 
