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plane A normally bisecting line 1,2 the locus (J/) consists of a curve 
lying in A and of the locus of the centres M,, AM. But the latter 
consists of two different curves, as the points J/,, M, never coincide 
during the rotation of A about / For a coincidence of M,, M, 
requires that the circles 234 and 134 coincide; as 1 and 2 are fixed 
points, this only happens when 1, 2, 3, 4 are coneyelie, but then 
the four points Mx belong to different branches of the locus. 
§ 3. The locus of the points M/,, M/, situated in A passes six times 
through the midpoint MZ, of 1,2, for the sphere on 1,2 as diameter 
cuts y* elsewhere in six points. So this locus is of order eight and 
will be indicated by wu’. 
The plane IF, at infinity contains the centres of four circles 
determined by the points of e* at infinity. The remaining four points 
common to IY, and pg, originate from two nodes generated as follows. 
If 4 touches y’, the point of contact is the pole of the line at 
infinity of .4 with respect to all the “circles” lying in that plane ; 
so M, and M, coincide then in that point of contact, but belong to 
different branches. 
Through / pass three planes containing four concyelie points; in 
the centre of each of the three corresponding circles 1234 the curve 
u° has a node. 
By assigning to J/, and M, respectively the points 4 and 3 we 
establish a correpondence (1,1) between the curves u* and of; so 
these curves have the same genus. As the singular points of a curve 
of genus one are equivalent to 20 nodes, the sixfold point J/, and 
the five nodes already obtained form the singular points of u*. So 
this curve is of rank sixteen; its four tangents through J/, originate 
from the four tangential planes of 0‘ through / in which planes M, 
and JM, coincide. 
$ 4. The locus of M, (and likewise that of W,) is a twisted sextic 
we’; its points at infinity are the points of g* at infinity and the 
points of contact of y’, with planes through /. 
Evidently it has five points in common with /; so it is rational 
and of rank ten. 
The three curves u“, u,°, u‚° concur in the centres of the circles 
lying in the three cyclic planes through /. Furthermore each curve 
w has still one point in common with u“. For in the plane A 
touching v° in 1 (or in 2), Jf, (M,) is at the same time one of 
the points. M,, M,; for 3=1 gives 124= 324 and therefore /, = M,. 
