Mathematics. — “On a Quartic Curve of Genus Two in which an 
Infinity of Configurations of Dusarcurs can be Inscribed.” 
By Prof. W. van per Wovpr. (Communicated by Prof. J.C. 
KLUYVER). 
(Communicated at the meeting of January 25, 1919). 
In an article entitled “The quartic Curve and its Inscribed Con- 
Jigurations’ H. Bateman’) comes to the conclusion that there exist 
quartic curves of genus two in which an infinity of configurations 
(10,, 10,) of Desaraurs can be inscribed. BATEMAN makes only 
mention of the existence of these curves without entering more 
deeply into their properties. 
Starting from considerations quite different from those of Bateman, 
I wish to indicate in this paper, what condition is sufficient for a 
uninodal quartic curve y, being circumscribed to an infinity of these 
configurations. It will appear that each point of y, is part of one 
of these configurations, and that we can construct each of them 
from one of its points if we consider y, as being given. I shall first 
mention a few known properties of an arbitrary uninodal quartic. 
1. Let for the present y, represent a quartie which has a double 
point in O and is for the rest arbitrary. We denote the tangents 
at O by z and y, their equations are c=U and y= 0; each of 
these two lines meets y, in One more point; the line joining these 
points is represented by z= 0. 
We can then represent y, by: 
7, = wy (@° + may + y? + 27) + 2 (av? + bay + cay? + dy*)=0. (1) 
Out of OU we can draw 6 tangents to y, if we do not count the 
two at O; the points of contact are the intersections situated out 
of O, of y, with the first polar curve of O, represented by 
a 2 wyz + ant + bay + cay’ 4+- dy? = 0. 
Hence 
Ya + Nrs 0 
indicates a pencil of quartics all of which possess a double point 
in O and touch « and y in that point; the other base points are 
h American Journal of Mathematics (36). H. Bareman. The quartic curve and 
ils inscribed configurations. 
