232 PEOFESSOR C. J. JOEY ON QLTATERNIONS AND PROJECTIVE GEOMETRY. 
the e(|uatio]i tlierefore imposes a single linear restriction on the line and represents 
a linear complex. 
In terms of vectors, putting q = I + p, p = 1 + cr, and using the expression 
given in the ‘ Elements’ (Art. 364, XII.) for a linear quaternion function, we have 
fq = 6 + 6 + Se’p + (jip, fq = a e Sep + fp'p ; 
fi/J = ^’u + A + + </>oP> f/l = G - ’ 
where 
Co = c, 2eo = e + e, 2q = e — e'; = (^o + Vp, <j)' = (f)Q — Yp, 
and tlie equations of the quadric and linear complex assume well-known forms 
-f 2Seop + Sp(/)op = 0, Sq(CT — p) + SpVptiT = 0 . . . (45). 
11. The equation of tlie polar plane of a point a with respect to the quadric 
(compare (42)) is 
Bqf,a = 0.(46), 
and /ort is the pole of this plane with respect to tlie unit sphere. 
Thus /o^ is the symbol of the polar plane of the point a,. 
With respect to the quadric the pole of the plane 
Sqh = 0 is 2 ^ = fo 
(47), 
and the reciprocal of the ([uadric has for its equation 
= 0 . 
The lines of the complex through a given point a lie in the plane 
Sqfa = 0 
while the point of concourse of the lines in the plane 
Sqh = 0 is 2 ^ = 
and 
(48). 
(«). 
(50) , 
(51) 
is the equation of the reciprocal of the complex. 
12. The nature of the united points of the function is easily ascertained. 
Since the function is the negative of its conjugate, its symholic quartic (22) 
must be of the form 
And if 
// + + Y = or (// - 6 i 2) (// - s/) = 0 . 
. (52). 
fPh = SiPi, f 2Y = -SjP'i./Ps = s,Pz,fp'o = - s.^p'z • • . (53), 
it follows in the first place (43) that the united points all lie on the unit sphere, and 
in the second by (27) that 
