PROFESSOR C. J. JOEY ON QUATERNIONS A^sD PROJECTIVE GEO.^IETRY. 
:73 
The com 
equation"' 
plex of lines connecting points and their 
(fPPM) = 0 
correspondents has for its 
. (29G); 
and the locus of points whose connectors to correspondents intersect a fixed line ah is ' 
the quadric surface 
{fqqah) = 0 .(297). 
The reciprocal of the complex (29G) is the complex of the conjugate 
{fPPfPl) = ^ .(298); 
for the line is reciprocal to the line 'qj, fp if "^pfp' - ^pf'q' ~ 0, 
which requires p', q', fp, fq' to be coplanar. 
the formulse ot this article comprise many theorems witli respect to the normals 
of confocal quadrics. It may also be observed that the complex (29G) i.s unchanged 
when /is replaced by (/+ x) (/+ yf. 
71. An arourary quadric has eight generators which connect a qooint and its 
correspondent in cm arhitrary transformation. This is the extension of IIautltox’s 
celebrated theory of the umbilical generators. (Compare Art. 40.) 
The conditions that the line q = fa -f- sa should be a generator of the arbitrary 
quadric surface 
SqFq = 0 .(299) 
are 
S«Fa = 0, Su(/T + F/)« = 0, Sc/'F/a = 0 . . . . (300) ; 
so that we can determine eight points a as the intersections of three known quadrics, 
and the lines joining these points to their correspondents are tlie common generators 
of the complex and the quadric. 
Four of these lines are generators of one system of the quadric and four of the 
other system. 
Four of the lines must belong to one system of generators. Let these be 
determined by the points a^, «o, a.,, rq. The condition that the line ptq should 
meet the line a^fa^ is 
(m/Q) = 0 or S (j)f/) [oj/oJ + S M (rq/rq) = 0 . . (301); 
and because any line vdiich meets three of these four lines likewise meets the fourth, 
we must have for proper selection of the weights 
(^b/b) + (fQ>o) + (a3/a3)-h(ajbj = 0, = 0 (302). 
where 
* When we refers and q to the united points of/, the equation of this complex takes the forms 
“ (b’b + GO) (f' - y'z) {xw' - z'w) = 0, E {t., - 4) {q - q) qjzx'iv + y'z’xw) = 0, 
p-=xa + yh + zc + wd, q = x’a-\- y'b + Fc + w'd. 
A \ector equation may also he employed, for if we put^i-l + a, q^p + p, the equation of the complex 
may be replaced by 
(/+/) p-M(/+s)(l+a), or p==r(V (/+q'‘i(l + a)-aS(/+/L’(1+a)), 
when we eliminate s by separating the scalar and vector parts after inversion of/+/. 
\ OL. CCI.—A. 2 N 
