PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOxMETRY. 237 
The symbolic equations now give 
F- = 0, G'^ = 0, Hp = 0, G=~Hf, F=np . . . (77). 
and 
Hq = xcG + yci + za where foT = d, fd = a, fa = 0 . . (78). 
The solution thus takes the more explicit form, 
n"'q = p era" + yd + za ; FIp = xd + = — xa, Fp = 0 . ('79), 
and z alone is arbitrary. 
If the last condition is not fulfilled, z is infinite. 
Again, where F" = 0, the solution is any point on the line, w variable, 
g = + ; p = xa"+ya'+zrt ; fp = yd-\-ya •, f ~p=-xa\ = 0 . (80). 
The symbolical equations satisfied hj F, G, II and /are now 
/A = 0, G^=0, H^ = 0, f^ = 0 .(81)_ 
Although the forms of the equations for F and G are identical, the nature of these 
functions are widely different; G reduces an arbitrary point to the line xa + ya, 
which is destroyed by a further application of the same function ; I reduces an 
arbitrary point at once to the point wa^ which is destroyed by a successive operation. 
The type of a function of this class 1^. is 
/_/(aaa"a"') = a {ctqd^d''^) + d [adqd^') + d' (adFq) . . . (82), 
in which a, a, a" and cf" are arbitrary quaternions. 
The function, 
f{q) . {adhb'') = a {aqhl^) + tf {adqh") + (6 + tjl) {aa hq) . . . (83) 
belongs to the sub-class I 5 . 
16. I/. A function of the second class destroys two points, a and />, and in virtue 
of the distributive property it destroys the line a, b. 
Since the locus of fq is a line (61), the function F vanishes identically (15), and 
likewise the invariant n' as well as n. 
Hamilton’s relations become, 
n = n' = 0 = 0, Gj = 0 ; Ilj -fi = F\ H F f — • • (^'^) 
and the symbolic equations for f and G degrade into 
F = P - -f F'J= 0 ; G {G - n") = 0 .... (85). 
The function G — n'^ destroys the line a, 5, which is consequently the locus of Gq. 
For the solution of the equation fq =p, the relations (84) give 
