1270 
Mathematics. — “Characteristic numbers for nets of algebraic 
surfaces.” By Professor JAN DE Vrins. 
(Communicated in the meeting of January 29, 1916). 
§ 1. We consider a general net of surfaces #”, with 7’ base- 
points B, represented by the equation 
n n n 4 
> Ay + Bbz + Yer =O. 
For a base-point 4, with coordinates 47, is 
n n na 
ay — 0, by == Okand cy, —0. 
By the substitution gv, = yz + den we find 
5 n—1 Ae Aen Ff n—1 n—1 
a(niay aztn Way,  az,+..+B(nrb, b2+...)+y(nde, et) = 0 
If the point Z is taken on a straight line, which has in } four 
coinciding points in common with a #7, we have 
u—1 n—l 
dy a; + Boy b:+ yey c:—0, 
n—2 2 n—2 ,2 1 —2 2 
aa, Gz+Bby 62+ yey ¢z=0, 
r—3 
aay, : a. + Bm b + yer Ee 0} 
Eliminating «, 3, y we obtain from this the /ocus of the tangents 
t,, which have in B with a ® a four-point contact; it is a cone of 
order six, which will be indicated by (¢,)°. 
If to the three equations just considered is associated the condition 
n—4 4 n-—4 ,4 n—4 4 
aa, a,+ Bb, bed ye, ¢:=0 
we have for the determination of the tangents with five-point contact 
in B 
| n—1 n—2 2 n—3 3 n—4 4 | 
| ay ay Cy az ay dz ay az | 
= n—1 n—2 ,2 nooo n—4 „4 | eed 
n= by b. b, bz b, b. b, be I= 0. 
| 
n—1 n—2 2 n—3 3 n—4 4 
| Cy Cz Cy Cz Cy Cz Cy ee | 
If the fourth and the third column are respectively cancelled, the 
equations thus obtained represent two surfaces’), of the orders 6 
and 7. To their intersection belong the straight lines which are 
obtained by equating to zero the matrix of the first two columns. 
The number of those straight lines apparently amounts to 3°?—2=7. 
1) The first surface is the cone (¢,)® already mentioned, the second has in B a 
sextuple point, is therefore a monoid. In order to see this, the substitution 
02k = Yk-+ eae may be done in the equation of the monoid; we then find 2° = 0. 
If a vertex of the tetrahedron of coordinates is laid in B the monoid is replaced 
by a cone. 
