270 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
((Q1Q2Q.3Q1.Q5)) — 0 .( 279 ) 
which is intended to denote that every set of four of the included ([uaternions is 
linearly connected. 
66 . For the numher of points common to the surfaces whose equations are 
obtained by deleting two of the (juaternions included in triple brackets 
(((Q1Q2Q3Q4Q5Q6))) = 0 .( 280 ), 
Salmon’s formula (‘ Modern Higher Algebra’) gives 
N = (281). 
67. To complete the scheme, we may regard the equation 
[[QiQaQ 3 QJ] =0 .(282), 
as requiring the four quaternions Qi, Qg, Qg, to be collinear; or the four curves 
(260), obtained by omitting one quaternion, to have common points. If these points 
exist they satisfy the equation (compare (279)) 
((^QiQaQsQQ) = 0 .(283), 
or lie on the complementary common to the five surfaces. 
A curve meets its complementary (‘ Geometry of Three Dimensions,’ Art. 346) in 
t = m{ix + V — 2) — r .( 284 ) 
points, and in particular for the curve [Q 1 Q 2 Q 3 ] and the two surfaces (wQiQ.^Qg) = 0, 
(Q1Q2Q3Q1) = 0, we find the number to be (compare ( 274 )) 
(285). 
These points are generally variable with the arbitrary quaternion «. 
Again, the surface 
(aQiQaQg) (6Q1Q3QJ + ^(aQiQjQJ (^QiQsQs) = 0 . . . . ( 286 ) 
inteisects (Q 1 Q. 2 Q 3 Q 4 ,) = 0 in [Q 1 Q 2 Q 3 ] = 0 , [QiQoQJ = 0 , and in the complementary 
corresponding to b. When we seek the intersection of the curve [Q 1 Q 2 Q 3 ] = 0 with 
its complex complementary on this surface, the number of points is found to he 
2 G + + MpMg -f~ -j- and these can all be accounted for by (285) and 
(259). 
We can also in this manner determine the points common to the two complemen- 
taiies(283) answering to a and b to be SM^M^Mg, employing the characteristics (278), 
and putting Mg = 0 . 
