1449 
Mathematics. — “Circles cutting a plane curve perpendicularly.” 1. 
By Prof. Hk. pr Vrins. 
(Communicated in the meeting of January 29, 1916). 
In the “Proceedings of the Royal Academy of Sciences at Amster- 
dam’’, section I, volume VIII, N°. 7, 1904, the present writer published 
a paper, entitled: “Anwendung der Cyklographie auf die Lehre von 
den ebenen Curven’’'), in which the circles are investigated cyclo- 
graphically, which either touch one or more plane curves once or 
several times or osculate them. 
At the end of that paper the observation is made that by means 
of a slight alteration in the plan, the circles may also be investigated 
that cut one or more plane curves once or several times perpen- 
dicularly; the aim of the following paper is to carry out that 
investigation. 
§ 1. As before we start from a plane curve 4” of order u, class vp, 
with d nodes, % cusps, t bitangents, « stationary tangents, and which 
moreover passes s-times through each of the two absolute points at 
infinity, and o times touches the straight line at infinity of its plane. 
In an arbitrary point P? of the curve we think the tangent ¢ to be 
drawn, and consider it as the locus of the centra of all the circles 
cutting the curve perpendicularly in P; if we then bring through ¢ 
the vertical plane (the plane 3 of the curve itself, the base, being 
supposed horizontal), and if we draw in it through / the two 
straight lines enclosing with ¢ angles of 45°, the cyclographic image 
circles of the points of those two straight lines are exactly the above 
mentioned circles cutting the curve /” perpendicularly. 
If we call the two 45°-lines 6, and if we repeat the construction 
indicated for all the points and tangents of the curve, all the straight 
lines 4 are the generatrices of a non-developable ruled surface 2, 
non-developable, because the two systems of circles cutting the curve 
perpendicularly in two infinitely near points, have no circle in 
common. That 2 is symmetrical in regard to the plane of the curve 
is to be seen at once, while no more proof need be given that the 
cone of direction is a cone of revolution with vertical axis, and 
whose generatrices with that axis enclose angles of 45°. This cone 
cuts the plane at infinity of space along a conic #,, which touches 
the absolute circle in the two absolute points /,,, /,, of the plane 
loo 2 oo 
B; for the point at infinity Z, of the axis of the cone of direction 
is the pole of the straight line at infinity /, of the plane of /”, as 
a 
1) Henceforth we shall quote this paper for the sake of brevity as “Anw. Cykl 
