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.