PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 259 
but oil the supposition that and Fo are self-conjugate, it follows (135) that these 
united points form a tetrahedron self-conjugate to the two quadrics SqFp^ = 0, 
SqFcq = 0. Take therefore any quadric to which this tetrahedron is self-conjugate ; 
F^ is determined and Fj follows from (207). 
Otherwise the function 
F. (q) . (ahcd) = xa (qhcd) + yU {aqcd) -f zc' {ahqd) + wd' {ahcq) . (209) 
is self-conjugate (Art. 21) when [ad/c'd') is the tetrahedron reciprocal to {abed) ; and 
on comparison with (208) the function F^ may be written down. The four scalars 
X, y, z, w are arbitrary, as might have been expected, since each self-conjugate 
function involves ten constants, while f involves sixteen. 
If two functions can be simultaneously reduced to the forms 
y; = F-iF„ ./o^F-iF^ 
( 210 ), 
the united points 
quadric, or 
In this case the 
of and /j must form tetrahedra self-conjugate to a common 
F«, — [biC^d{], &c. Fa.o = .(211)- 
eight united points are so related that any quadric 
SqFgq = 0 
( 212 ) 
which passes through seven, passes also through the eightli. 
The condition that the point should be on the quadric may lie written (211) 
Sf/iFgF-i = 0, or (F-iFgCq, c„ d,) = 0 . . . (213), 
and if tq, and d^ are likewise on the quadric, it follows (Art. 24) that the first 
invariant of the function F'^Fg vanishes. Hence if the points a.,, b.,, c.^ are also on 
the quadric, the remaining point d^ must lie on the quadric too."^' Thus one of the 
united points is fixed with respect to the others, and the functions J\ and /o must 
satisfy three conditions, which reduce the number of their constants to 29, and this 
is precisely the number involved in the two quotients F“T^, 
* Compare Appendix to the New Edition of the ‘ Elements of Quaternions,’ vol. ii., p. 364. 
