1109 
any plane perpendicular on « contains eight poles. o° is four times 
intersected on d°, by the cone of revolution w? (§ 3); the remaining 
intersections are poles of six principal circles. The principal circles 
form therefore a system with index six. 
As a base-point B is vertex of three 8’, consequently lies on three 
principal circles, these circles are to be counted twice. 
5. We shall now consider the system formed by the osculating 
circles, y,, of the conics of the pencil. For a point R‚ of as 
consists of /_ and the asymptote touching in R,. To the osculating 
circles of the two parabolae belong the figures consisting of /, and 
a diameter. 
The asymptotes envelop a curve of class three,*8, which has 1, as 
bitangent. The tangents of °3 passing through a base-point B are 
apparently the lines connecting 4 with the other three base-points. 
The circles y, passing through a point P and a point Q) consist 
of 7, combined with an asymptote or a diameter of a parabola; 
their number amounts therefore to jive. 
From this it may be deduced that through any two points P,Q, 
five osculating circles may be laid. 
First it may be observed that the locus of the centres of the circles 
y, passing through P must be a curve c°, for five of those circles 
have their centre on /,. 
If the system [y,| is considered as the cyclographic representation 
of a surface 2, c° is the orthogonal projection of a curve o'° lying 
on 2. The latter has 20 points in common with the orthogonal 
hyperbola uw? ($ 2) determined by P and Q. Of these 10 lie on the 
curve d°, (§ 2) representing the asymptotes; the remaining 10 form 
5 pairs of poles of circles passing through 7? and Q. Consequently 
Jive circles 3 pass through two given points. 
The cone of revolution w’? ($ 3) with vertex P has in common 
with @ the curves Jd’, and o'°; so @ is a surface of order six, 
Hence, any point of the plane vr is centre of three osculating circles. 
6. Let S be the intersection of a 8? with the osculating circle 
which has B as point of contact, w a ray passing through & parallel 
to one of the axes of 3’. In order to investigate how often a straight 
line & drawn through 4 becomes chord of osculation, we associate 
the reflected image s of & with regard to « to the straight line f, 
touching §? in 5. To a line ¢ belong two lines u, but only one 
line s; a ray s determines with / two lines wu, but only one p’, 
consequently one ray ¢. The two coincidences s—t belong to two 
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