( 750 ) 
We finally put the question how many twisted circles can be 
brought through five points where we understand by a twisted circle 
a twisted cubic, cutting the isotropic circle in two points. All twisted 
cubics through these five points describe on the plane at infinity an 
involution; if now a point describes the isotropic circle, its conjugate 
points will describe a curve of order eight having with this circle 
sixteen points in common; four of these points are at the same time 
double points of the involution, whilst the other lie two by two on 
a same twisted circle. 
So through Jive given points pass ten twisted circles , of which four 
touch the plane at infinity. 
Mathematics. — “On the surfaces the asymptotic lines of which 
can he determined by quadratures”. By J. Bruin. (Com¬ 
municated by Prof. Hk. de Vries). 
In a paper entitled as above A. Buhl (Nouv. Ann. de Math., 
4 e serie, vol. 8, page 433, vol. 9, page 337, Rev. sem. XVII 2, page 62, 
XVIII 1, page 58) discusses the surfaces given by the parameter 
representation 
<p{z) = aO + F{r) r 
in which x, y, z refer to a rectangular system of coordinates, so that z, 
8, and r are the so-called cylindric coordinates; these are the only 
ones which are used in the course of the investigation. 
Buhl now gives the differential equation of the asymptotic lines 
of 0 (z) = a 6 -|- F (r) with 0 and r as independent variables as well 
as with 2 and 8. It is then evident that this equation embraces 
many special cases, where the determination of the asymptotic lines 
comes to quadratures. 
We can put the question more in general: which are the surfaces 
of one of the forms 2 = <f (r, 8), or 6 — f(r, z), or r=f{z,8), whose 
asymptotic lines can be determined by quadratures? 
Starting from the differential equation of the asymptotic lines 
Ddu * -f 2Vdudv + jy'dv* = 0 
(Bianchi-Lukat, “Vorlesungen fiber Bifferentialgeometrie”, page 109), 
where D, D and D" have the values, to be found on page 87 of 
the quoted work, we find for the differential equation in r and 6 
of the asymptotic lines of z = <p (r, 8 ): 
51 
Proceedings Royal Acad. Amsterdam. Vol. XU. 
