1107 
Mathematics. — “Systems of circles determined by a pencil of 
conics’. By Professor JAN pr Vrins. 
(Communicated in the meeting ef Jan. 30, 1915.) 
The osculating circles and the bitangent circles of the comics of 
a pencil form two doubly infinite systems: of these I shall consider 
some properties in this paper. 
1. Any straight line # passing through a base-point B of a 
pencil of conies (67), is a normal line of one 3°. To » l associate the 
diameter 1 of 8’ passing through 5. As each line passing through 
B bears the centra of two 8’, a correspondence (2,1) exists between 
m and n. Each coincidence is an axis; each base-point ts therefore 
verten of three conics: the axes envelop a curve of class three, *a. 
As the line at infinity /, is axis for the two parabolae of the 
pencil, consequently ditangent of *e, only one axis belongs to a 
pencil of parallel rays. 
The axes a form on the rational curve *e@ a quadratic involution, 
of which each pair consists of the axes a,,a, of a definite conic. 
The central conic of the pencil (locus of the centra) is the curve of 
involution and at the same time part of the orthoptic line of °a; 
the missing part’) is apparently the line /,. 
The locus of the vertices 7’ of the conics 8? has a triple point 
in each base-point. As an arbitrary p° has four vertices, it will have 
16 points in common with (7); the curve in question is therefore 
of order 8. It has apparently nodes in the nodes D of the degenerated 
conics. The vertices of the conics le therefore on a (T)* with four 
triple points and three nodes. 
A. Hach g° possesses two systems of bitangent circles, 72,2. For the 
parabolae one system exists of the pairs of lines formed by a tangent 
and the line /,. 
As each point P bears three axes, P is the centre of three circles 
yoo. A perpendicular to the plane + of (8) contains therefore six 
poles of circles yap, in other words the system [2,2 | is the cyclographic 
representation of a surface of order six, w’. 
The intersection of wm’ with rt is apparently the locus of the foci 
(focal curve of the pencil); the latter is consequently a bicircular 
curve of order six, having the nodes of the three pairs of lines as nodes. 
The tangents p of the two parabolae are the images of the points 
at infinity on the cones of revolution mw’, the generatrices of which 
1) On a straight line 7, the pairs of orthogonal tangents of %x determine a (3,3). 
Any intersection of two orthogonal tangents is a double coincidence; the orthoptic 
line is therefore a figure of order 5. 
. me 
(3 
Proceedings Royal Acad. Amsterdam, Vol. XVII. 
