97 
the same direction. The result here found is, however, more striking, 
for, as the declinations were determined in exactly the same way 
for faint and for bright stars, the greater value for Z (the constant 
term in Ad) which the former give, cannot be ascribed to constant 
errors of the declinations of the catalogues used. If systernatic errors 
of the catalogues are to be made responsible for our result, it can 
only be the consequence of residual magnitude-errors in declination. 
This point certainly deserves further investigation. Another point 
that has not been investigated so far is the possible presence in 
the differences KistNer—Zonecatalogues of terms dependent upon 
multiples of «. 
Mathematics. — “Pencils of twisted cubics on a eubie surface”. 
By Prof. Jan pr VRIES. 
(Communicated in the meeting of March 25, 1916). 
1. The straight lines of a bisextupel of a cubic surface &* will 
be indicated in the usual way by a, and 4,; the remaining straight 
lines by cj/. In order to arrive at the wellknown representation of 
®’ on a plane tr, we lay rt through the straight line c,, and consider 
bb, 
of ®* is then represented by the intersection P/, on r, of the ray 
passing through P. The intersections A,, A, of 6,, 6, represent «,, 
a,, Whereas «a, a, @;, d, are represented by their intersections 
A, A,, A;, A,. The representation of the straight line bp is the 
conic 87, which is determined by the five cardinal points A; U F4); 
the straight line cj, is represented by Aj Ay. From this representation 
it may be deduced that any twisted cubie 9? lying on ®* has a 
sextuple as chords and is not intersected by the associated sextuple. 
2. A o® having the sextuple bj as bisecants is represented by a 
straight line of rt; a plane pencil with vertex C” is therefore the 
image of a system of o° all passing through the point C. Such a 
system we shall call a pencil; C we call the singular point of the 
pencil (9°). All o° rest on the 15 straight lines cj; and have the 
straight lines hj. as chords’). 
To (9*) belong six degenerated figures. For the straight line C’ A; 
N 
1) In my paper “A simply infinite system of twisted cubics” (These Proceedings 
Vol. XVIII p. 1464) I arrived at the consideration of such a pencil in an entirely 
different way. 
- 
Proceedings Royal Acad. Amsterdam. Vol. XIX 
as directrices of a bilinear congruence of rays. Any point P . 
