‘)--0 
— / -J 
PliOFESSOii C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
The equation of the complex may also be regarded as representing the assemblage 
of lines converted by Q into conjugate lines with respect to the unit sphere. 
G9. In order to determine the four lines common to the quadric and the linear 
complex, observe that the point of contact (/o“Vt) of a 23lane Shq = 0 with the 
quadric must also be the point of concourse (//“V;) of the lines of the complex in that 
jjlane, in order that the plane may contain lines common to the two assemblages. 
Therefore the points e in which the joairs of common lines intersect satisfy the 
equations 
e or h =/]c = uf^^e .(290). 
Thus four points e are determined, the united points of the function /o“Vj. 
It ajjpears, as in Aif. 12, that the latent roots of this function are eciual and 
ojjjiosite, and that the united points form a quadrdateral on the quadric. 
Otherwise, the invariants of and of are identical (Art. 23), and these 
funetions satisfy the same symbolic quartic ; and because their conjugates, 
and likewise satisfy the same quartic, it nmst be of the form 
{fo~vy + + n = o, or - u,^) - ui) = o . (291). 
Hence the lines in question are determined on solution of a quadratic equation. 
When these four points cq, e.,, cb are taken as points of reference,* so that 
„ „ + V<^\ I zrq + ?rcb _ + y'e\ 
+ 
' I / / 
AO c 3 
(292) 
the equations of the quadric and conqdex may by the aid of (290) (compare again 
Art. 12) be reduced to tlie forms 
xy + zw = 0 tq {xi/ — x'lj) + u, [zw' — z'tc) =: 0 . . . . (293). 
70. The locus of j^oints whose correspondents are in 2 )erspective with a fixed 
point a is the twisted cubic 
fq + tq - a or [fq, p, n] = 0 .(294), 
and the locus of lines which pass through a fixed point a and connect a point and its 
correspondent is the cone 
fq + tq = xfa + ya or {fqqfaa) — 0 .(295). 
* Observe that these four points c are the only points for which 
h E J'<1 E /o2 E //i> 
the signs = being iised to denote equality when the quaternions are multiplied l)y a suitable factor. For 
vector functions 
= ^'p = hP 
only when p = e, ■where « is the spin-vector. 
