238 PEOFESSOE C. J. JOEY ON QUATEENIONS AND PEOJECTIVE GEOMETEY. 
n"(i = Hp Gq-, Gp = 0 .(86); 
and since Gq may be any point on the line a, h, the locus of q is the plane [ify, a, li]. 
If Gp ~ 0 is not satisfied, the solution is an arbitrary point on the line a, h 
afiected with an infinite weight. 
11 3 . If i f ■= 0 , the solution is 
n'"q = p + Hq ; lip Gq= 0, Gp ~ 0 .(87), 
and 
= 0, HP = 0, Ilf= -G .(88), 
whence 
Hq = xd + ya + zh, Hp — xa if d =fa.... (89). 
11 4 . If further, 71 '" = 0 , the solution is 
q = xd'' + yd + za + wh, p — xd + ya, fp = xa, f~p — 0. , (90); 
and the general function of this type is 
fq {abdd') = a (ahqd') -fi d [ahdq) .(91), 
and the function G of is of this sub-class. 
17. nil- The third class is that in which f destroys three points a, h, c, which are 
not situated on a common line 
Here 
n = n' = n" = 0; F = G = Hf= 0 ; rf' f + H, P - n'J= 0 . (92), 
and the solution is 
n"'q = p + -f ^6 + where Hp = 0 .(93)- 
III 2 . If d" = 0, 
q = xd fi- + 2:6 + 'Wc where p) = xa, fd = a . . . ( 94 ). 
The type of the function is 
f{q) . (abed) = a (abeq) .( 95 ), 
to which the function F of I 4 belongs. 
18. IV. The fourth class is that in which two lines ab and cd are destroyed. 
n = d = 0, F=0, Gf= 0, Hf+ G = if = H +/= n"' . (96) 
and the symbolic equations are 
/(/-K') = 0 ; G{G-if) = 0 .(97). 
The function 
fid) • i<^bcd) = tpabqd) + tqd [abeq) .(98) 
is of this type. _/ destroys the line a, b and reduces an arbitrary jJoint to c, d; 
J — destroys c, d and reduces an arbitrary point to ab. 
