PEOFESSOE C. J. JOEY OX QUATEENIOXS AND PEOJECTIVE GEOMETEY. 231 
It is also possible to obtain relations connecting pairs of the points ((N), p. 224), 
= + 
; (^ 1 ^ 3 ) = 
( V 37 !•) 
(7374)__ . 
(7i737s74) ’ 
from which we learn that 
_ [ 73 y_ 
{71737374) 
- ^ = (71737374) ( 7 'i 7 V/ 37 ' 4 ) 
(34 
(35); 
and we are at liberty to write separately on further selection of the weights (for tlie 
products of the weights alone have been assigned), 
( 71737374 ) = (7'i 7'37'37'4) = \/ - f.(36), 
with corresponding simplifications in the formulm. 
When the function is self-conjugate, the tetrahedron of united points is self- 
reciprocal to the unit sphere. 
9. Introducing two new linear functions defined by the equations 
/ = /o-|-/;./=yo-/. or Va=.f+f’ \f, =/-/' 
(37), 
it is obvious that for any two quaternions, ]) and p, 
or symbolically 
(38) , 
(39) , 
and^Q is self-conjugate, and/) is the negative of its conjugate. 
10. The equation 
S9/„<? = 0. (40) 
is the general equation of a quadric surface, and 
Sr7/,p = 0.(41) 
is that of a linear complex, p and q being both variable points. 
In fact (40) is the most general scalar quadriitic function homogeneous in q, and 
the surface represented meets the arbitrary line q — a th in the points deter¬ 
mined liy the roots of the quadratic 
Sa/gCi + 2 ^Su/q6 + t^Shfol) = 0.(42). 
In like manner (41) is the most general scalar function linear in two quaternions 
and combinatorial with respect to both, for by (38) 
S7//7 = 0.(43) 
whatever quaternion q may he. It is therefore immaterial if we replace q and p in 
(41) by any other points on their line, provided the two points are not coincident, and 
