276 PROFESSOE C. J. JOEY OY QUATERNIONS AND PROJECTIVE GEOMETRY. 
Dimensions.’ The cone has h — m 2 double edges, and consequently three trijDle 
chords pass through an arbitrary point on the sextic. The sextic is thus a triple 
curve on the regulus of triple chords, and the surface has no other multiple line. 
76. There is yet another quadratic complex of importance in the study of a pah’ of 
functions. A point a is transformed into ocf^a + yf\a by the operation of + yf„, 
and as x and y vary, the locus of the transformed point is a line v’hich we shall call 
the satellite of a. 
The satellites generate the complex 
= 0.(315), 
and the form of this equation should be compared with (296) and (298). There is 
also the complex of conjugate satellites obtained by replacing and /o by their 
conjugates, but when the functions are self-conjugate, or when one is the conjugate 
of the other, the two complexes combine into one. For the functions /’and /’' this is 
= ^ .( 316 ). 
'T'he four fa, fa, f^a, frt form a harmonic range on the satellite of the point a. 
There are also harmonic properties connecting pencils of planes Sq/h = 0, Sp/'a = 0, 
Sp/’yi = 0, Sq /’« = 0 ; and it may be verified that these four jdanes intersect in a 
satellite for the inverse functions. This we shall prove for the general case. 
The reciprocal of the complex of satellites is the conqjlex of the conjugate satellites 
for the inverse functions. 
If p and q are any two points on the reciprocal of the satellite of a, 
Sy/ja = '^pfct = 0, = 0.(311), 
and on taking conjugates we see that the four points ffp, fp, f'q, fjq are 
co-nlanar, so that 
(//A/a'p/iV/Di) = 0.(318). 
The locus of points wdiose satellites meet the line ab is the quadric surface 
(compare (297)) 
= 0.(319). 
77. The satellite of a point which describes a line (j- = a fi- th constructs one system 
of generators of the quadric 
= (./i + ff) {ct -f th) .(320), 
but the regulus degrades into a system of lines enveloping a conic whenever 
(/i«/At/lW'^) = 0.(321), 
that is, whenever the line belongs to the reciprocal of the complex of conjugate 
satellites (318). 
The conic is co-planar with the line when the further conditions 
{alfafb) = 0, (abfaff) = 0 .(322), 
are satisfied (compare (296), (298)). 
