1°64 
The locus of the quartic (A)* corresponding to the different points 
A of a‘ is a surface which may be indicated by A. The curves 
of = (67, c*) passing through a given point B of @° lie on a c? 
meeting «* in eight points A; so B lies on eight curves (A)*, i.e. 
6’ is an eightfold curve of A and the same result holds for y*. A 
quadric 6? meets a* in eight points A and contains therefore eight 
curves (A)!; moreover it has with A the eightfold curve 8‘ in common. 
We conclude from this that A is a surface of order 32. 
§ 2. The lines joining two points P, P’ belonging to the same 
group of 7* form a complex I; we are going to determine its order. 
The curves of = (0’,c’) generate a bilinear congruence’). Any 
line is chord of one of; the points Q, Q’ determined on the lines m 
through M by the e* with m as chord lie on a surface (Q)° with 
M as threefold point; the tangential cone in M projects the o* passing 
though M. 
The two surfaces a® passing through Q and Q’ cut m in two 
other points R, A’. The locus (f) of the points A,’ has in M a 
sevenfold point, any plane u through M cutting (Q)* in a curve u’ 
with threefold point M/ and the surface a’ through M/ ina conic u’; 
so the seven points Q common to « and u’ and differing from M 
bring seven points # in M. So (f) is a surface of order nine with 
sevenfold point M. 
The curve @* common to (/) and u cutsu’ ind x d5—7 K 38=24 
points S differing from M, which can be arranged into two groups. 
In any point of the first group MS is touched by an a’. So these 
points lie on the polar surface J/* of Al witb respect to the pencil 
(a*)*). Consequently the first group counts 3 X 5—38= 12 points. 
In any point S of the second group a point A coincides with a 
point Q’; then the point Q coincides with A’ in a second point S 
and both points S lie on the same a’; so these points are associated 
and belong to the same group of /*. So the plane u contains six 
pairs P,P’ collinear with J/; in other words: the pairs of points 
of the involution 1* lie on the rays of a complex of order six. 
§ 8. The complex cone of M contains the seven rays joining M 
to the points MZ’ belonging with MZ to the same group of /*. So 
1) We have treated this congruence in a paper “A bilinear congruence of twisted 
quartics of the first species”, These Proceedings, vol. XIV, p. 255. 
2) The polar surface of (y) with respect to a?, + aw?,=0 is generated by 
means of this pencil and the pencil of planes a, qa, ia’, a’, = 0; so it is repre- 
sented by iy U? == My 4, Wa 
