50 DEMONSTRATION OF A THEOREM RELATING [112
112]
and I multiply as if a Q , b 0 , c 0 ... really existed, taking care to multiply without making
any transposition in the order inter se of two symbols a 0 , b a combined in the way of mul
tiplication. This gives a quasi-product
system
serving,
the prof
ww, + (aw, + a,w) a 0 + (bw, + b,w) b Q + ...
+ aa,a Q 2 + bb,b Q 2 + ...
+ ab / a 0 b 0 + a / bb 0 a 0 +....
Suj:
being ai
Suppose, now, that a quasi-equation, such as
the trip'
er = + ,
and sup]
means that in the expression of the quasi-product
b 0 c 0 , c 0 a 0 , a 0 b 0 , cJb Q , a Q c 0 , b 0 a 0
are to be replaced by
a 0 , b Q , c 0> — a Q , —b n , — c 0 ;
where e,
and that a quasi-equation, such as a 0 b 0 c 0 = —, means that in the expression of the quasi
product
b 0 c 0 , c 0 a 0 , a 0 b o , c Q b Q , a 0 c 0 , b 0 a 0
contain
are to be replaced by
- a Q , —b a , — c 0 , a Q , b Q , c 0 .
and ej M
It is in the first place clear that the quasi-equation, a Q b 0 c 0 = +, may be written in
any one of the six forms
«0 & oC o = +> K c o a o = +> c 0 a 0 b 0 = +,
«oCo 6 o = -> G 0 b 0 a 0 = ~, b 0 a o c 0 = — ;
and so for the quasi-equation a 0 b o c o = —. This being premised, if we form a system of
quasi-equations, such as
a 0 b 0 c 0 = ±, a 0 d 0 e 0 = ±, &c.
and by t
ively, we
where the system of triplets contains each duad once, and once only, and the arbitrary
signs are chosen at pleasure ; if, moreover, in the expression of the quasi-product we
replace a 0 2 , 6 0 2 ,... each by —1, it is clear that the quasi-product will assume the form
Also wj
®« + fl A + M 0 + CA+ ....
w //> *//> K> G „ ••• being determinate functions of w, a, b, c, w,, a„ b„ c, ..., homo
geneous of the first order in the quantities of each set ; the value of w„ being obviously
in every case
w a = ww, — aa, — bb, — cc,...,
whence i
and a„, b //} c,,,... containing in every case the terms aw. + a/w, bw,+b t w, cw, + c,w,... but
the form of the remaining terms depending as well on the triplets entering into the
