573 
1907-8.] Algebra after Hamilton, or Multenions, §11. 
We now introduce a generalisation of the conjugate 0' of a linity 0. 
[Read 0' as “ 0 dash ” and 0' as “ 0 shad ”]. Let v be a given linity such 
that 
v' = v _1 = ± v 
and let 0' = v0V _1 
[When 0 is identified with a multenion q, then Rg where R is any 
uniretroplacement becomes a case of 0' ; and R 2 where R is any unipro- 
placement becomes a case of r0r -1 .] 
0' will be called the ^-conjugate of 0. 0 is said to be i/-self -conjugate 
when 0' = 0; y-skew when0'= — 0; y-rotational when 0'0 = 1, and there- 
fore 00' = 1 . 
i/-conjugacy is almost identical in properties with conjugacy, which, of 
course, is a particular case given by v— 1. Thus 
(0')' H 0, (00)' = 0'0\ (0'- 1 /-0'~ 1 . . . • (31) 
Also 0' has precisely the same n-tio, as 0 ; because r0V -1 has the same n - tic 
as 0', and <p' the same ^-tic as 0. Also since (0 M ) V = (0') M when M is an 
integer, positive or negative ; we have with all the meanings of / considered 
above 
/(0') = [/(0)]' (32) 
In particular : — the g, and £(a + ^ 0 ) of 0' corresponding to the root a 
[which occurs A times in the 0 n - tic and g times in the minimum degree 
identity of 0 ; and therefore occurs A times in the 0' ^-tic and g times in 
the minimum degree identity of 0'] are £ 0 ' and £'(a-|-£ 0 ') respectively. 
The statement in square brackets is true, because whenever /(0) = 0, then 
/(0 X ) = 0 ; and whenever /(0 V ) = 0, then /(0) = 0. 
(32) shows that when 0 is ^/-self-conjugate, then also is/(0). When 0 
is i/-skew ; /(0) is i/-self-conjugate when / is an even function and is r-skew 
when / is an odd function. 
More generally, the i/-self -conjugate part of /(0) is J[/(0) +/(0')] an d 
the j/-skew part is J[/(0) — /(^')]- When 0 is i/-skew these parts become 
the even and odd parts of /(0), that is J[/(0) d=/( — 0)] ; and when 0 is 
j/-self -conjugate they become the whole and zero respectively. 
The general form of a r-rotational function is e x where y is any i/-skew 
linity ; for, in the first place, e x is i/-rotational since 
(ex)' = ex' = e~ x = (ex) -1 : 
and in the second place, when 0'0 = 1 
log o 0' + lo go0 = ° 
[never unintelligible when 0'0 = 1 since this gives that the (discriminant) 2 
. (30). 
