236 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
Ij. One latent root is zero. In this case 
n = 0, Fp = 0 ; n'q = Gp + Fq = (Yy? + xa.(68), 
because F reduces every quaternion to the fixed quaternion a multiplied by a scalar x. 
Here X is arbitrary, provided the condition Fp = 0 is satisfied ; the point Gp lies in 
the fixed jjlane (58) ; and q may be any j)oint on the line joining this point to a. or 
in other words, this line is the solution of the equation (67). 
If the condition Fp = 0 is not satisfied, the scalar x must be infinite, so that in 
the limit f {Gp + xa) may have a component at the point a, which escapes 
destruction by F. The solution is simply the point a attected with an infinite 
weight. 
Wheji n — 0, it appears from Hamilton’s relations that F satisfies the depressed 
equation 
F {F - n') = 0 ..(69), 
and the interjiretation is, F reduces an arbitrary quaternion to « ; F — F destroys a. 
lo. Two latent roots are zero. Here 
n = n' = 0 , Fp) = 0 ; Gp + Fq — Q ; n"q = Hp + Gq . . (70), 
and q must lie allowed the full extent of arliitrariness consistent with the conditions. 
Observing that the relations (56) now give 
pG = 0, GP = Q .(71) 
it appears that the double operation of f destroys the result of operating on any 
quaternion by G^ and that G destroys Hence, 
Geq = xa + ya, where fd = a^ f'^d =0 .( 12 ). 
The scalar x is determinate for 
fGq = Gp = xa .. (73), 
lint y is arbitrary, and the solution is any point on the line, y variable, 
n”q = Hp + xd + . ..(7I)- 
As before, if Fp) is not zero, the solution is a multiplied liy an infinite scalar. 
The character of tlie function G has now conqiletely changed. It now destroys 
a line {f ^q), and because Gf ~ — 0 , or Gf '^H^ — 0 , and also n' = 0 , the symbolic 
equations of G and F are both degraded, and are 
G{G-n"f = Q, i7^=0 .(75). 
I 3 . The solution in this case is 
n = n' ■= n” = 0 ; Fp — 0, Gp + Fq = 0 , Hp Gq — 0 \ n"'q = + Hq . (76). 
