1903-4.] Dr Muir on General Determinants. 75 
to prove that the result was correct. As the mode is that in which 
the rule of signs is dependent on the number of interchanges,* or, 
as Boole calls them, “binary permutations,” any interest attaching 
to the little exposition is connected with the “ proof.” The first 
essential paragraph is : — 
“The result of the elimination of the variables from the 
equations 
a l x l 
+ 
a 2 x 2 
+ • « 
, . + 
= o, 
Vi 
+ 
b 2 x 2 
+ • • 
.. + 
: * 
= 0, 
Vl 
+ 
+ 
• * + 
^ rP^n 
= 0, 
is an equation of which the second member is 0, and of 
which the first member is formed from the coefficient of 
x x x 2 - • • x n in the product of the given equations, by assum- 
ing a particular term, as a 1 b 2 • • *r n , positive, and applying to 
every other term a change of sign for every binary permutation 
which it may exhibit, when compared with the proposed 
term a x b 2 - • -r n . 
The curious point worth noting here is that we are directed first 
to form the terms of the expression afterwards denoted by 
+ + 
\a 1 b 2 • • • r n \ and called a “permanent,” and then to alter the 
signs of certain terms of it. Boole then proceeds : — 
“ The truth of the above theorem is shown by the following 
considerations. The elimination of x x from the first and 
second equation of the system introduces terms of the form 
a x b 2 -a 2 b ly a x b z — a 2 b x , etc., in which the law of binary 
permutation is apparent, and as we may begin the process of 
elimination with any variable and with any pair of equations, 
the law is universal. From the same instance it is evident 
that no proposed suffix can occur twice in a given term, 
which condition is also characteristic of the coefficient of 
x x x 2 . . . x n in the product of the equations of the system, 
whence the theorem is manifest.” 
See Rothe’s paper of the year 1800. 
