234 PEOFESSOR C. J. -TOLY ON QUATERNIONS AND PROJECTIVE GEOW.TRY. 
II. The second class consists of functions having a line locus of united points, 
with the following sub-classes ;— 
II^, the two remaining united points distinct. 
IIo, the two remaining united points coincident. 
II3, one of the remaining united points on the line locus. 
11.. , the two remaining points coincident and on the line locus. 
III. The third class consists of functions having a plane locus of united points, and 
there are two sub-classes :— 
IIIj, the remaining united jDoint is not in the plane. 
111., the remaining united point is in the plane. 
IV. The functions of the fourth class have two line loci of united points. 
It is to be noticed that any peculiarity in a function is exactly reproduced in 
its conjugate. This will appear clearly from the following discussion, but the 
proposition is virtually proved in the concluding remarks of Art. 6, 
To assist in the examination of the different cases, it is convenient to repeat 
Hamilton’s relations (20) and (21), and in addition to obtain the symbolic cjuartics 
for the function 11, G, and F. These quartics are deducilile from the relations (20) or 
(21) without much trouble. The group of formulm is thus :— 
Ff=n, FFOf^n', G-\-Hf=n\ H+f=n’'; 
H = F" -f G = n" - F'fFf \ F = n' - n"f+ rF'P - P ; 
fi _ + n'[p — A/+ n = 0.(56). 
IF - + {u" + 3A"2) IP + {n - H 
+ — n7i"' -|- — 0. 
G^ - 2n"G^ + {2n + FiP + n"^) G- - {2nn" - + 7 P + iFFu'") G 
F' — nn'n" -h n'~n" = 0. 
F^ - + n'F'F'^ - nhi"'F+ P = 0. 
14. For the sake of brevity in discussing the various classes, one root of the scalar 
quartic is supposed to be reduced to zero by rejilacing the function by one of the four 
functions f— Q,/— Q, f— fs,/— hi. of Art. 6 ; and wlienever there is a multiple 
root, it is the multiple root which is reduced to zero. 
I. One qur.iternio^i, “ o,” is reduced to zero hy the operation of the function. 
Ttememljering that the conjugate also reduces a quaternion a' to zero, it follows if 
/n = 0,/V=0 .(57) 
that the locus of the transformed points, p> = fep is a fixed plane. 
Spa' =0.(58), 
