1108 
intersect the plane r at angles of 45°. As two tangents p may be 
drawn in any direction, the curve at infinity d? of those cones is 
nodal curve of @’. 
The circles passing through two points ?.Q are the images of 
an orthogonal hyperbola wu’, situated in the normal plane in the 
middle of PQ. with the middle of PQ as centre, while its asymp- 
totes intersect + at angles of 45°. 
As it has four points in common with d*, it intersects @* in eight 
finite points, whieh will be the poles of four circles yo. passing 
through PQ. Through two arbitrary points pass therefore four 
hitangent circles. 
If one of these points is a point at infinity, each of the four 
circles is formed by /, with one of the tangents going from the 
other point to the two parabolae *). 
3. The circles y22 passing through a point P form the image of 
the intersection of w° with the cone of revolution w’, having P as 
vertex, while its edges intersect t at angles of 45°. The latter has 
in common with w°, 
besides the nodal curve d*_ , also a 0°, containing 
the poles of the circles yo9 passing through Z. Hence: the locus of 
the centres of the bitangent circles passing through a fired point ts 
a curve of order four, pr 
The tangents from to the two parabolae determine the points 
at infinity of this curve. It is intersected by the perpendicular at 
the middle point of PC in the centres of the yo. passing through 
Pand: 6 
Let us consider the corresponding loeus for the case that P is 
replaced by a base-point B. Any ray n passing through B is normal 
line of one 6°, consequently contains the centres of two yo. touching 
this 8? in B; B cannot be centre of such a circle; so the locus in 
question is a conic. This was to be expected, for the four tangents 
of the parabolae determining the centres at infinity coincide here in 
pairs. The central curve p* will have the perpendicular in the middle 
of PB as bitangent. 
4. The circles having the axes of g? as diameters belong to the 
system [yo]. These principal circles are represented on w° by a 
twisted curve o°, having the central conic of (3°) as projection. For 
any point of the centre is centre of two principal circles so that 
1) Similar considerations concerning the system of the orthoptic circles of (6?) 
may be found in my paper ‘On the orthoptic circles belonging to linear systems 
of conics” (These Proceedings I, 305—310). 
