PROFESSOE C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 229 
He then equates the coefficients of the arbitrary scalar t in the symbolical equation 
n,=f]F, = F,j] .. . (19), 
and obtains the symbolical equations 
n = Fj\ n' = F-{- Gj\ n" = 6 ^ + Hf, F" = H+f 
m, 
which will be found to be of great importance in the 
geometrical theory. 
In virtue of (19), all these functions are commutative, 
in order of operation. 
These equations establish certain colliiieations which 
are illustrated in the annexed figure. 
From the relations (20) Hamilton deduces 
H = n'" ~f; G = F' - F"f-\- p ; F = F - 
and the symbolic quartic satisfied by / 
fi _ n"'P + 71"P - Ff+ n = 0 or (/ - p (/- p (/ - t,) (/- p = 0 . (22), 
if h, P P roots of the quartic 
ti _ n'" F + 7i"F - Ft + n = 0.(23), 
or the latent roots of the function f. 
It appears by (12) and (14) that exactly similar equations are valid for the con¬ 
jugate function /', it being only necessary to replace F, G and H liy their conjugates 
F\ G' and H', as the invariants n, F, r/'and n"'' are the same in both cases. 
7 . The united points of the transformation are represented Ijy the quaternions 
qi, ^3 and which satisfy the equations 
PFi — Ph ; Ph = kF .; Ph = kF'PP .’ 
and they are determined by operating oir an arbitrary quaternion liy the function 
obtained by omitting one factor of the second form of (22). In like mannei by 
omitting two or three factors of the same quartic, the equations of the lines joining 
two, and of the planes through three, of the united points are obtained by operating 
on a variable quaternion. Thus 
9 = if- k) if- ip and q = (/- pr .(25) 
are respectively the ecpiation of the line through the points q^ and of the plane 
through the points q^, q^, q^. These results are obvious when the aiffiitrarily variable 
point is referred to the united points as points of reference, or when we write 
r = x^q^-^ -\-x^q^-\- x^q^ .(26). 
