PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 277 
But when e aiG givGu, as liGro, a seriGS of taiigGiits to a conic lionioffraphic with 
a series of points on a line in its plane, in three cases a tangent passes through its 
corresponding point; and evidently when a point lies on its satellite, it also lies on 
the sextic = 0 ; so the line under discussion is a triple chord of the sextic. 
It seems worth while noticing (compare Art. 66) the remarkahle equation 
(((«> ^/hh/A/Vd/o^))) = 0 .(323) 
of the cisfiernhlage of triple chords oj the sextic, for this equation is equivalent to (321) 
and (322). 
78. Again, in an arbitrary plane = it is generally possible to find one point p 
whose satellite lies in the jilane. The conditions are 
^lp = 0, S/Aj;= 0, so p = [l,f:i, ffi]. . . (324); 
and the point is determinate unless the reciprocal of the plane lies on the conjugate 
sextic (Art. 73), or, in other words, unless the plane is a united jilane for some 
function of the system. In this case (compare (312)) there exists a line locus for 
points p whose satellites lie in the plane. 
This is precisely the case of the last article, so when the envelope of satellites is a 
conic co-planar with the line, the plane is a united plane. 
79. For an arbitrary plane, the locus of points whose deriveds by -|- xf\ remain 
in the plane is the line of intersection of (y\ + xf„) q = 0 or (// + ocf') l={) 
with the given ^ilane S/q = 0. All these lines pass through the point p, which may 
be called the focus of the jdane. 
Assuming an arbitrary point p to be a focus, the plane of which it is the focus is 
(compare (324)) the reciprocal of the point 
^ .(325). 
The relation between a focus and the reciprocal of the plane is of the same nature 
as the correspondence discussed in Section XIX. (conqmre (526) with (324)). 
The points whose satellites pass throug'h a given point a lie on a twisted cubic 
= 0, 
and the locus of points whose satellites lie in a plane is a right line. The satellite 
of a point q and the plane ^Iq = 0 pierces the plane in the point 
A -MWil .(326), 
and from this quadratic transformation connecting the points q and q^, it follows 
that q (or q) describes a conic when (or q) describes a right line. In the former 
case the conics joass through the focus of the plane. Thus again an arbitrary line 
qq' meets the satellites of two points on the line (compare (320)). 
It would take too long to explain the various geometrical relations in the plane 
