ON THE THEORY OF DETERMINANTS.
Suppose 17 = 2 (p!; + crv + •••) (gmj+ o!x' 4- ...)
this expression being the abbreviation of
U = (p ^ + arj + ...) (ax + a'x + ...) + ....
(p/f + <*■/*? + ...) (a,a? + a,V +...) +
+
[(ft — 1) lines, or a smaller number].
KU= 2a p, 2a o-,...
2ap, 2aV,
is = 0
which follows from the equation (®).
Conversely, whenever KU=0, U is of the above form.
Also FU = — a#+aV+..., rx + bV+ ...,.,
R^+Sy + ..., Xap , 2a a ,
r'£ + s'r) + ..., 2ap , 2a , er ,
which may be transformed into
(for shortness, I omit the demonstration of this equation).
And similarly,
UU =
[12
(48),
(49),
(50),
.(51),
k.X + A V ..., BX + B V ... , ...
llf + S rj..., n'g+s'r) ... , ...
P > O’
a , a' ,
(52),
RX + R V + ..., Sa? + s V + ...,...
Af+B?7 + ..., A'f + B'r) + ..., ...
b
Q_
a , a' ,
...(53),
where it is obvious that if the sum 2 contain fewer than (ft —1) terms, FU= 0, 717=0.
The equations (52), (53) express the theorem, that whenever KU = 0, the functions
FU, UU are each of them the product of two determinants.