534 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Hence [since 2/3S( )|« is a general form] if 0 is self- conjugate and equal 
to 2/3S( )|a, it is also equal to 2aS( )\/3, or to half the sum of these two. 
Hence we have the following: any fictorcolinity (self -conjugate fictorlinity) 
is given by 
# = J5(aSp|/3 + /3Sp|a) (9) 
If we put 0 for the self -conjugate (or colinity) part ^(0 + 0'), as it is 
called, of 0, 
<pp = ^(aSp\fi + f3Sp\a) when 0p = 3/?S/o|a . . . (10) 
If we put (p s for the skew part J(0 — 0 '), as it is called, of 0, 
^spM 2 ~ a Sp|/5 + @Sp\a) = -h'2S r (S 2 P a )\P • • (11) 
[(U) § 4], 
Let L(y, e ) be any function (scalar, fictor, multenion, etc.) linear in each 
of its constituent fictors y, e. Then with (8) 
L(£,0f) = SL(a,/?); (12) 
always L(£, 0f) = L(0'£, £) ; (13) 
with a colinity 7 always 
ULH) = HH,0; • • • • (14) 
with a skew linity always 
L &</>£) M-m&O; • 
with (8) L (£, «) = L(«, £) = £S[L(a, 0) + L(ft a)] 
(15) 
116) 
If x is an arbitrary fictorlinity and x an arbitrary fictorcolinity, then if 
0 and i/r are fictorlinities, 
0 = 0 when Sx£|0£ = Sx £| H • • • • (17) 
0 = 0 when Sx^!0^ = Sx^|0C • • • • (18) 
[For (17) put x = /3S( )|a and for (18) x = |[/3S( )|a + aS( )|/3] .] 
We will here make a digression to prove that when Law A holds, f and 
y are invariants, that is, that we get the same meanings for them if in 
their definitions the fictits q^q -1 , qi 2 q~ l , • • • • (where q is a fictor product) 
be used in place of q, i 2 , ... . 
Let </> = q( )q~ Y , so that by (6) 0' = q~\ )q = <j>~ 1 - [It is here that Law A 
is assumed. By (6) 
Soj0'p = Sp|0o- = Sgo-g _1 |p = S(cr^ _1 |p.g) = Sq^pqlo- 
because if Law A holds | <j = i 2 ar, | p = i 2 p. If Law A were not assumed to hold 
we should get 0' = Kg( )Kg _1 in place of 0 / = g~ 1 ( )q\ just as if 0 be a 
multilinity given by (pr = qrq ~ 1 , its conjugate 0' is given by 0V = Kg.r.Kg -1 .] 
By (13) 
h(4>i,^) = U4>4>'i,0 = m,0- 
