PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 239 
19 . As the theory of the self-conjugate linear vector function clilfers in various 
details from that of the self-conjugate c[uaternion function, it is necessary to devote 
a few remarks to the latter. 
The four united points of a self-conjugate function form a tetrahedron self¬ 
conjugate to the unit sphere, for in this case the two tetrahedra of Art. 8 coincide. 
If two united points coincide, they must coincide with a point on the spliere, and the 
scalar quartic has a pair of equal I’oots. But in the case of a real self-conjugate 
vector function when two latent roots are equal, the function has an infinite number 
of axes in a certain plane, and not a single axis resulting from the coalescence of a 
pair ; and the reason is simply that a real vector cannot l:)e perpendicular to itself, 
while each axis of a self-conjugate vector function must be perpendicular to two 
others. For a quaternion function, on the other hand, a real point may he its own 
conjugate with respect to the unit sphere, and there may be in this case 
coincidence of united points without a locus of united points and consequent 
degradation of the symbolic quartic. 
Again, the roots and axes of a self-conjugate vector function must he real, because two 
conjugate imaginary vectors, a + \/ —1 a — — 1 yS, cannot be at right angles 
to one another, since the condition is — 0, while a“ + is essentially 
negative. But two united points of a real self-conjugate quaternion function may lie 
conjugate imaginaries, the condition 
S (a -f y - Ih) {a — v/ - l/>) = -f- S?)® = 0 . . . ( 99 ), 
merely showing that the real points a and h are situated one inside and one outside 
the unit sphere. 
20 . On account of the importance of the self-conjugate function, it may not he 
superfluous to illustrate cases of coalesced united points. 
Writing for the general self-conjugate function, 
/(I-h p) = e + € + Sep + cpp ; S (1 + p)/(I -f p) = e-f 2Sep + Spfl>p .(100), 
the latent quartic is 
—f" (e + mf') + [em' + m — e®)— t [em -j- in + Se (fl> — m") e) 
-f m (e ~ Secfl-i g) == q.(101 ). 
The quadric surface S^/q = 0 has its centre at the extremity of the vector — flwQ, 
or say at the point c. 
One root is zero if 
e — Sefl)-^ 6 = 0 .(102), 
and the quadric is a cone with its vertex at the point c. A second root is zero if 
m = — Se (fl> — m" + e = — ?uSe<I)“^e, or if T<I)“h =1 . . ( 103 ); 
that is, if the vertex is on the unit sphere. 
