555 
1907-8.] Algebra after Hamilton, or Multenions, § 9. 
namely [(3) § 3] ( — ) 4n(n_1, (i 2 ) n . From (10) we clearly have the following 
fully extended forms : 
/ «odd N R2 = teo±?2 + 24±26+ ) 1 
Vcr 2 =±i; +»*%! ±2 3 + 2g ±g 7 + . . . .) f 
/ n even \ R? = (q 0 + q 2 + g 4 + q 6 + - • • .) ( 
\<£ ,2 = + 1 / + q 3 + + ^7 + • . • •) i 
( 11 ) 
( 12 ) 
Here x is understood to be +1 or —1. xxz is put instead of vs to enable 
us to pass from one m to another. 
(11) and (12) may be supposed to give the general form of any 
complementary proplacement and RK any complementary retroplacement. 
There is a very important difference of property between (11) and (12). 
Even order multenions alone occur on the right of (11), whereas Rg in (12) 
is a perfectly general multenion. 
When n is odd a complementary replacement reduces the multiplex of 
order n to one with the properties of the continent multiplex. 
Let now [(17) § 3] P 0 = J(l + P), P x = J(l — P) and define B q by 
B q = (P 0 + (13) 
It will be found that B is the value of R when we give x a special value, 
thus 
R = B when x = ( t 2 )* (w+1) (n+2) . . . . ( 13 a) 
This statement may be verified by finding from (13) the values of B q 0 , Bg p 
B q 2 , Bg 3 . At first B will be looked upon as a multilinity whose pro- 
perties are to be investigated. 
By examination of the four cases n — 0, 1, 2, 3 it is easy to show that 
invariably 
K n -*(i+w 2 ) = K i(w y z )( n - 2 (14) 
It is convenient to put 
rc-i(l+c? 2 ) = m (15) 
By means of [(5)] us{ )t< 7 _1 = P” -1 and PP X — — P 1 it is easy to see the 
effect of transposing K with tz¥ 1 thus 
K^tsrP^) = KP^.Ksi = 
= cr 1 P n - 1 P 1 Kg 
or K^SXPjg) = ( - ) ,l_ 1 trT 2 .wP 1 (Kg) .... ( 16 ) 
To transpose K ??l with then we have to multiply by [( — ) n-1 Ttf 2 ] m . The 
value of this is easily found by use of the relations established by putting 
iu 2 = ±l 
(or 2 )* 1 -©*) = sr 2 = ( - l^-tJJ = l, ( - i)^+^ 2 > = - cr 2 . (17) 
