PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE (4EOMETRY. 271 
SECTION XII. 
On the Geometrical Relations depending on Two Functions and on the 
Four Functions /, /', and /^. 
Art. 
68 . Relations of the quadric Sqfoq = 0 and the complex = 0 to the linear transformation 
produced by the function /. 
69. The quadrilateral common to the quadric and the linear complex. 
70. The quadratic complex of connectors of points and their correspondents. 
71. The extension of Habiilton’.s theory of the “umbilicar generatrices”. 
72. The locus of the united points of functions of the system {x'f+y’f' + z')~^ i^f+yf ' + ^) 
is the twisted sextic [fq, f'q, ?] = 0. 
73. The case in which / and /' are replaced by arbitrary funetions. The sextic intersects a 
united plane of the system in three residual collinear points. 
74. The numerical characteristics of the sextic. 
75. The surface generated by its triple chords. 
76. The satellite q = xfiO. + yf. 2 a of a point a, and the quadratic complex of satellites . . . 
77. Satellites and triple chords of the sextic. 
78. An arbitrary plane contains one point and its satellite. 
79. The/ocMS of a plane. Case of a united plane. 
80. Special case for the functions f and /' . . .. 
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68 . We devote this section to the study of the geometrical relations connecting 
a function / with its conjugate /', its self-conjugate part and its non-conjugate 
part (Art. 9), and to the relations connecting a pair of arbitrary functions 
and /g. 
The quadric 
^qfq = Sqf'q = Sqf^q .(287) 
is the locus of a point which is conjugate with respect to the unit sphere to its 
correspondent in each of the transformations due to/, /' and 
The linear complex 
= 0, or Spfq = 8qfq), or Spfq = Sqfp .... (288), 
may be written in the form (compare p. 223). 
Spq'S/q = Sqp'S/p, (pSp = p, p'S/p =/p).(289), 
which expresses that the product of the perpendiculars from q', the derived of one 
point Q, and from the centre of reciprocation on the polar plane of another point P 
with respect to the unit sphere, multiplied by the perpendicular (S/q) from q on the 
plane which is projected to infinity by the transformation, is equal to the correspond¬ 
ing product of three perpendiculars found by interchanging p and Q. This property 
is also true when f is replaced by its conjugate f'. 
