579 
1907-8.] Algebra after Hamilton, or Multenions, §11. 
and we note that the replacement A which is to be identified with 
conjugacy in the linity sense is 
A = .... (43) 
It may be called the alternate replacement or retroplacement because 
alternate fictits are negatived. Similarly, when e v e 2 ... . in (37) or 
X 1 ,\ 2 . . . . in (41) are taken as fictits we may call the arrangement an 
alternate fictit arrangement. 
If with our new, fictits i v i 2 , .... we adopt law A, we have 
Law A 
(44) 
We now have when q is a multenion of the multiplex i 1 i 2 .... i 2m : — 
(A) A q the alternate replacement of q is the conjugate of q in the linity 
sense. [It is obvious that q may be identified with a linity in a multitude 
of ways. With one of those ways A q is the linity conjugate.] 
(B) Therefore any uniretroplacement of q which [since I 1 = TJ7q( )i 1 ~ 1 'U5~ r \ 
may always be expressed as vA. qv~ x is a ^-conjugate of q in the sense applied 
to Unities above. And the two species of ^-conjugate above, depending on 
whether v is + v or — v , depend now on whether Av is + v or — v. 
(C) From what was said above about the root sum of the multilinity 
v( )v it follows that the root sum of q is 2 m Sg. Changing q to q c we get 
that the sum of the c th powers of the roots of the identity of degree 2 m 
which q satisfies is 2 Sg c . 
We may now transfer to multenions the theorems proved above for 
Unities. Thus ignoring (24), (25), every equation from (15) to (32) may be 
taken as a multenion equation by reading 
v, q , r, A q, = vA^v -1 , R( ) or “ R-conjugate ” 
for v, <p, i {/, <p', “ v-con jugate.” 
We can now prove the statement made in the footnote of § 4 above. 
We have to show that if qpq 0 = 0 for all multenion values of p, belonging 
to a given even order multiplex, then one of the multenions q, q 0 is zero ; 
or, what is the same since such a multenion is a linity, restricted to a given 
order of Unities ; if = 9 lor all linity values of y, then one of the 
Unities (p , \js is zero. Take the Unities to be fictorlinities. If \[r is not zero, 
there is at least one non-evanescent fictor 3 = \Jsa obtained by operating by 
\[s finite fictor a. By proper choice of y, /3 may be converted into any 
fictor = y required. Hence, that ^y \]s may be zero, <p must kill every 
fictor y whatever ; i.e. when \[s is not zero, (p is zero. 
