1907-8.] Algebra after Hamilton, or Mnltenions, § 6. 531 
Putting here 
S ft a (c " 1, a (n - c) a = ( - ) n_c S n a^ ) aa (w - c) = ( - ) n ~ c ^ 
and transposing the £ that remains with S n _ c a (w-C) we obtain 
S^KaKafc?) = a£r c fl) f = Ka<? 
or taking conjugates 
S c (d< c ipa) = 4 C) . 
This establishes complete reciprocation between the 2 W independent 
multenions a? based on the n fictors a_i, a_ 2 , .... and the 2” independent 
multenions a {c \ based on the n fictors d_ lt a_ 2 , .... Thus corresponding to 
(7) we have 
(15) 
Putting c = n in (7) or (8) or (15) we have 
af)Kair>=l (16) 
which expresses the reciprocal of a given determinant in terms of it. 
Moreover, the 2 n relations (8) express similar symmetrical properties of the 
minors of any order of the two determinants. 
Let p be an independent variable fictor belonging to the multiplex of 
order n and V the differential operator defined by 
P = 'Six, ^ = '2 lD x (17) 
Thus if q is a function of p, 
dq = SdpK^.q (18) 
and if /3 be any fictor, 
P=^ 1 S Pl Kp = K^ 1 Sp i p (19) 
Let L(/3, y) be any function of /3, y linear in each. Then 
L( Vl , Pl ) = ^L(i,D xP ) = ^L(c, 0 .... (20) 
so that Vi and pi may be interchanged. Define f 2 , f 2 , etc. by the 
equations 
L(2, C) = L(Vij Pi)} £ 2 ) = L(vp Pi? V 2 ? P 2 ) • • (21) 
(19) becomes 
P=&£Kp = XZStP (22) 
By putting one f = 2aSAKJ=2aSaKf we get 
L(£, £) = 2L(a, a) = SLffi, a) . . . . . (23) 
Applying this to each pair of f ’s in 
ScfA .... S(S e f 1 J 2 .... t)Kq=m& 
