290 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
because the first conjugate of the first conjugate of f{'pq) is simply the function/(p^) 
itself 
When the successive accents are difierent, the laws connecting the various functions 
are deducible from the relations (compare (414)) 
(’’V) = {/Thp) = (/'). {<P') 
= iP') = S'/ if')^ M = Sp (/■")' {(jr) 
= Sr/, (qp) = Sq (/)' (rp) = Sp (/)" (qr) 
(^ 16 ). 
in which q>, q and r are perfectly arbitrary. 
These relatiojis show that 
i/TiP-l) = (/"),(W) = (/)' (P'j) =f"('U>): 
and thus any multiply accented function may be reduced to one or other of six 
fundamental functions, the function and its two conpigates and the permutates of 
these three functions. 
103. Exactly as in Arts. 5 and 6 , the equations 
(/(«'i) - ; /(c'i) - to ; f{dq) - td) 
= (/' {oiq) — ta ; f (bq) — th ; /' {cq) - tc ; f [dq] - td) (418), 
(/(pa) - ta ; f{i>h) — th ; /(pc) — tc ; /(pd) - td) 
= {r{po<) - tn ; riph) - th ; /" (pc) - /c ; ^ {pd) - td) (419) 
are identities for all quaternions q, a, h, c and d, and for every value of the scalar t. 
The first is obtained on the supposition that f{pq) is a function of and the second 
on tlie supposition tliat it is a function of q. Dividing each member of the identities 
l)y [ahcd), we obtain the Inquadratics 
Jid) - tJ'iq) + thd"{q) - tJ'"{q) + d, 
Hp) - tV (p) + tH" (p) - tP" (p) + d.(420); 
and J (q), J' ( 7 ), J” ( 7 ), J"' ( 7 ), of the fourth, third, second and first order respectively 
in 7 , are the Invariants of f{q>q) considered as a iunction of p. Equating these 
biquadratics to zero, we obtain the equations whose roots are the latent roots of 
f{Pd) ^ function of 7 .) and as a function of 7 . 
It is evident from (418) and (419) that tliese relations are equivalent when the 
function Is permutahle, and then I[q] = '^( 7 ), &c. 
104. The quartic surfaces 
(4-1) 
