2Gf; PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
coincide with those originally found, hut in a dilferent order. In this case Y is the 
conjugate of X, and the scheme (251) becomes 
(XX') (X'-iX-i) (XX-i) (X'-^X') 
(XX') (X'-^X-^) (X'-^X') (X X-i) .... (252). 
In this case the conjugate of a transformed function is the transformed function of 
the conjugate. 
Again, in the general case, when,Y = X~i, the types of the upper row (251) 
merge in the single type (XX“^), and the conjugates in the type (X'^^X'). 
Finally, all types unite in the single class (XX') when X is the inverse of its 
conjugate (Art. 47). 
GI. Covariant functions may he derived by the following general process, as well 
as by multiplication and division. For arbitrary scalars, Q, &c., 
nt{ttf)-\ahc\ = {ttfa, ttfb, ttf'c] = F,,^[_ahc-] . . . (253), 
where Uf is the fourth invariant of %tf, and where 
Fn^Whc\ = t[f\a,f'J),f^c'\ .(254), 
the summation in this last equation referring to permutation of the suffixes. 
These functions belong to the (Y~^X~^) class, because 
= . . . (255), 
Ux and 7?y being the fourth invariants of X and Y. 
In like manner functions of the (XT) class are obtainable in the form 
/i23[«?>c] = S[Ai'rt,FV5,A3'c] ; .Fi'[rt5c] = [/a,/i6,/ic] . . . (256). 
G2. Although rather foreign to the subject of this paper, it may be as well to indicate 
the nature of the Hamiltonian quaternion invariants of a system of functions. It 
was stated in a paper on Quaternion Arrays* that these invariants are included in 
the quotient 
^ fi(^' /a ' 
/ 2 « U 
> -h- (ahccl) . 
(257), 
fJ) ffi fud 
a h c d 
formed by dividing a four-column ai'ray by {abed), each row of the array consisting of 
the residts of operation by a single function on four a.i’bitrary quaternions. Briefly, 
* ‘ Trans. Roy. Insh Acad.,’ vol. 32, p. 30. 
