268 PROFESSOR C. J. JOEY OX QUATERNIONS AND PROJECTIVE GEOMETRY. 
but as it is clesiral>le to determine also its rank and the number of its apparent double 
points, we shall adopt a different method. 
The cpiaternions a and h being arbitrary, the identity 
Qi + Qa (Qs^^Qi) + Qs + «(^-^QiQA) + ^ (QiQcQa^O = o • (-gi), 
shows that the two surfaces 
(aQiQ,Q,) = 0, OQ,Q.Q3) = 0 .(262) 
intersect in the curve (2GO), and also in a complementary curve common to the three 
Sl-II*fRC0S 
(a&Q3Q3) = 0, (aiQ3Q,) = 0, (cAQ.Q.) = 0 .... (263); 
for when (262) is satisfied, the identity shows that either (260) or (263) must be 
satisfied. 
Let m denote the order of the curve (260) ; then the order of the complementary is 
(Ml + Mo + Mg)'^ — m = rrJ .(-6t), 
the orders of the two surfaces (262) being M^ + M^ -}- Mg. 
Again, considering the intersection of the second and third surfaces (263), it follows 
from the identity that they intersect in the complementary curve and in the new 
curve 
[Qff?>] = 0 . (265); 
and because the orders of the surfaces are Mj + Mg and M^ + Mo, the order of 
this new curve is connected with m' by the relation 
(Ml + Mo) (Ml + Mg) - m' = .(266). 
Again, writing down tl)e identity 
a (hcqCl^) + I (c^iQio) + c ('iQpff^) + q (Qi«^t^) + Qi {ahcq) = 0 . (267), 
in which q is the variable cpiaternion, wliile o, h and c are constants, it appears 
exactly as before that the surfiices 
(o7>f/Qi) = 0, (a5cQi) = 0 .(268), 
of orders Mi + 1 and Mi, intersect in the curve (265) and in a complementary curve 
which is obviously the comjdete intersection of the surfaces 
[ahcq) = 0, (n6cQi) = 0 . (269); 
that is, a |ilane and a surface of order Mi. 
Now the relations* 
* 8almox’s ‘Geometiy of Three Dimengions,’ Arts. 345, 346. 
