PROFESSOE C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOiMETRY. 235 
because = 0. Every plane through the point a is reduced to a line ; every line 
through the j^oint becomes a point; the scalar n is zero ; the function F reduces 
every point to a and destroys every point in the fixed plane (58). 
The qnadrinomial (3) must reduce to a trinomial, for/cannot destroy a quaternion 
unless there is a relation between rq, cq, or else between bo, /q, h^. The 
type of functions of this kind is 
fq = + agSc/gq + cigSc/q ; « = a' = [aiU/q] . . (59). 
II. The function destroys two distinct points. 
If 
fa = 0,fb = 0 ; f'o! = 0, f’b' = 0.(60) 
the line a,-6 is destroyed. The locus of the transformed points is the line of inter¬ 
section of the planes 
Spa' = 0, Sp)b' = 0.(61). 
Every plane and every line through the line a, h is reduced to a point. The 
function is reducible to the binomial type 
fq — af)a\q -f a.fta’yq ; a th = \a\a\r'\ a' + t'b' = [a^ap*] .... (62), 
when r and r'are quite arbitrary, and it is evident (15) that the function F vanishes 
identically. 
III. The function destroys three non-collinectr points. 
/a = 0,/b = 0, yc = 0 ; fa' = 0,f'b' = 0,f'c = 0 . . . . (63) ; 
and every point is reduced to a fixed point, the intersection of the planes 
S^ja' = 0, Spb' = 0, S 2 :>c'= 0, or p = [a'bV] .... (64). 
Hence the function is a monomial, 
fq = [a'b'c'] S [cibcfi = afa'^q .(65), 
and the function G vanishes identically. 
IV. The function destroys two distinct points, a and b, and alters the iceights of two 
others, c and d, in the same ratio, but otherwise leaves these points unchanged. 
The type is 
fq . {abed) = tfabqd) + tgd{abcq) .(66). 
15. In order to illustrate the nature of the solution of the equation 
f<l=T .(67) 
in the different cases, we employ Hamilton’s relations (56), which give the solution 
on substitution. 
2 H 2 
