30G PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
are transformed from the points r ^ \/5 r '; these latter are real if s is positive : 
otherwise they are imaginary. 
To discriminate between the outer and the inner region on the line (500), observe 
that the vectors from the centre of reciprocation to the limiting points are 
^ V/-(rr) _ V/(rV) 
^ S/(r,-)' P S/(,-V) 
(501); 
and that the vector to the point is 
_ V/’ [rr) + sYfjr'r) ^ p^f {rr) + sp^ f {rV) 
S/(rr) + 6'S/(r'/) S/(7 t) + sS/(rV) ’ ' ' 
The point p lies on the inner region if S/(rr) and s^f{r'r') are of like sign; and 
the inner region corresponds to real points if the points r and r' are either both 
inside or both outside the quadric 
s/ [qq] = 0.(503). 
This quadric is the locus of points projected to infinity; it may of course be 
imaginaiy, so that S/ (rr) and S/ (r'r') are essentially one-signed if r and r are real. 
In this case the region is always inner. If the quadric is real, the points r and r' 
(if real) cannot both he inside, for they are conjugate to it. The nature of the 
intersection of a line with this quadric controls the nature of the conic into which 
it is transformed. 
134. The locus of the Jacobian correspondents of points in a plane is a sextic 
curve, and for the peimutable function this sextic cuts an ailiitrary plane in points 
which correspond in pairs. There are therefore three connectors in a plane. 
Ihe vertices oj the triangle of connectors belong to the same octacl; for if pi is one 
vertex and po and pg the points, one on each of the connectors through p,, which 
(Art. 132) belong to the same octad as pj, then r/o and pg lielong to a common octad, 
and their line is a connector—the third connector in the plane. 
We may suppose the weights of the points p. and pg chosen so that the 
Jacobian correspondents are 
q -2 ± q-i, P's dz Pn Pi db po.(504), 
the vertices of the triangle being (Art. 132) harmonically conjugate to these points 
in pairs. 
135. Let the eight quaternions which represent points of an octad have their 
weights chosen so thaU^ 
ho -/(piPi) =/(p3p,0 = =/(psPs).(505), 
It follcjws from Art. 132, that this convention is the same as that made at the end of the last article. 
