272 
Through each point of y’ passes a plane pencil of trisecants; this 
appears moreover from (5): the plane pencil in the plane (A) has as 
vertex the intersection of the planes a’, + 2a", =0, 6’, + 76", = 0, 
Cea ae == 
§ 2. As the system (1) may be replaced by the system 
ua?y + Bb’ + yer, 0 
aa’, + Bog = POL GN 4 Lime bar WE ee (8) 
aay + BD! + ye", === tl \ 
all the curves 0° lie on the quartic surface ®*, represented by 
" " " 
a a b Lr Ca 
Through a point of ®* passes, in general, one curve gy’; we shall 
therefore call the system (9°) a pened. 
An arbitrary straight line is therefore cut by for curves 0%. On 
N 
3 
®* lies also the curve 7’; any trisecant ¢ intersects d* on y* and 
in the three points, in which it meets the corresponding curve 9°. 
All the 9° pass through the points C, and C, indicated by ¢,? — 0, 
c', =0, c's =0. These points are therefore singular points of (9°). 
From (1) it appears that the surface ®* may be produced by 
combining the pencil 
a@(a40'5—= Cant AU ey CO a} 0. 
with one of the pencils 
a (a?o — era's) + Bb — 0775's) = 0} 
NC RE SA 
As product of two projective pencils we find then besides #* the 
(10) 
plane. ¢’, = 0 or «the ‘planese;. 0: 
In connection with this we consider the curve ¢*, in which 7% 
is intersected by the arbitrary plane g, as product of a cubic pencil 
with a quadratic pencil. The first, (7°), has two base-points /,,F, on 
the intersection f of g with c’;=0 and seven base-points / (k= 3 
to 9) on g*. The second pencil, (v7), has a base-point G, on f, the 
remaining three, G;.(4 = 2,3,4) on g*. The projectivity has been 
arranged in such a way that two homologous curves intersect in a 
point Q of f. 
The two pencils determine on g* the same involution /*; each 
group Q,(k—=1 to 5) consists of the intersections of ~ with one 
of the. ¢urvesio®. 
