1130 
curve A of Jacopi, which is of order 3(n—1). The third group 
consists of (18 n—-33) points, where a c” has four consecutive points 
in common with its tangent. From this it ensues that the paints of 
undulation of a net form a curve of order (18 n—33). *) 
The curve (P) is of order 3 (n—1) and has a triple point in P; 
through P pass consequently (9n*—21 7) of its tangents. They now 
form two groups: the first consists of base tangents f, the second 
of tangents wu in points of undulation. 
(P) now intersects the curve A in 3 (n—1)(2n—8) points D, 
of which one of the two tangents passes through P (class of the 
curve of ZEUTHEN)*), consequently in 9 (m—1)’? — 3 (n—1) (2 n—3) 
or 38n(n—1) points D, for which the base tangent ¢ passes through P. 
From this it then ensues, that P lies on (6n?—18n) tangents 
u. The four-point tangents, therefore, envelop a curve of class 
6 n (n—3). *) 
Mathematics. “On a Representation of the Plane Field of Circles 
on Point-Space’. By Dr. K. W. Warsrra. (Communicated by 
Prof. JAN DE VRIES). 
(Communicated in the meeting of January 27, 1917). 
$ 1. The circles in the plane YOY are represented by 
CX 4 Y* — 2aX — 2bY¥Y He=0. 
If we consider a, b, and c as the co-ordinates wv, y,z of a point, 
a correspondence (1, 1) is obtained between the circles of a plane 
and the points of space. The image of a circle is obtained by 
placing a perpendicular in the centre on the plane and by taking 
on it as co-ordinate the power of the point O with regard to the 
circle. 
For the radius we have 7? = a? + 6? —c. 
Cireles with equal radii are therefore represented by the points 
of a paraboloid of revolution, with equation #? + y? —z=7". 
The images of the point-circles lie on the limiting surface G, 
a + y* = 2, 
a paraboloid of revolution, touching the plane NOY in O. 
§ 2. A pencil of circles is indicated by C, + 4C,;=0. For the 
circle à we have 
1) Another deduction of this number is to be found in my paper: “Character- 
istic numbers for nets of algebraic curves”. (These Proceedings XVII, 937). 
2) Cf. my paper ‘On nets of algebraic plane curves”. (These Proc. VII, 633). 
5) These Proc. XVII, 936, ; ; 
