274 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
Hence the eight points common to three of the quadrics 
== 0 {n=l,2, 3, or 4).(303) 
are likewise common to the fourth. But four of these points are the united points of 
the function f, while the remaining four determine (297) four lines of the complex 
(296) which meet the four generators. These four lines are common to the quadric 
and the complex, and make up with the other four the complete system of eight lines. 
In accordance with (302) we may write for the two sets of four lines* 
+ {chfih) + = 0 . 
{n\fa\}-\-{a'„Ja\2}-\-{a’^fa'^}-\-\a\fa'^] = Q .... (304), 
and it maybe remarked that a direct interpretation of (302) is that four equilibrating 
forces can be placed along the lines of either set, for the first equation (302) expressed 
that the resultant of four forces vanishes, and the second requmes their moment with 
resj)ect to the centre of reciprocation to he zerof (see (33), p. 230). 
7 2. The locus of the united points of all fanctions of the system 
-^y'f + ^') M*/+ ?//'+2=).(305) 
is the curve 
= 0 . . . . _.(306); 
and this curve (276) is a sextic whose rank is 16, and the number of whose apparent 
double jDoints is 7. 
If q is a united point of a function (305) and t the corresponding latent root, we 
obviously have 
{x — tx^)fq -h (?/ — ti/)f'q + (z — tz') q = 0 . . . . (307), 
whence (306) follows immediately. 
The sextic curve is evidently the locus of united points of the conjugates 
{x f-\- V/’+ 2 ) of functions xf yf + 2 , but it is not the locus of united points of 
conjugates of functions of the general type (305). 
In the following articles we shall consider some part of the theory of two arbitrary 
functions and as it is partially applical^le to the subject under discussion. 
73. The loci of the united points of all functions of the two systems 
«A + y'fz + Wi H- yfi + 2 ) and (.</■/ + y'f.' + z')-^ (.rfi/ + yfj + z) (308) 
are respectively the sextic curves 
Lfyifm'] = 0. = o.(309)- 
These two curves unite in the special case of/q = ./’/. The first is the locus of the 
united points of the system xf-^ + y/ij fi- z, and the second is the corresponding locus 
for the conjugate system. 
* Iti the notation of arrays 2 { 'Pn<ln} = 0 implies 2 {pnQn) = - [T’™?'!] = 6- 
t If an = AnSiin, fiin — BiiSfan, A,i = 1 + a.„, B,i= 1 +/S,,, the equations (302) become 
2 ilSn-a-n) Sa„S/fl,i = 0; 2Va„/?„Sa„S/«n = 0. 
