238 
PEOPESSOE BOOLE ON THE THEOET OF PEOBABILITLES. 
principal elements, viz. in A;„— XA^i and A„j— XAi,-. 
proved, 
M^=LN; 
Then, by what has already been 
but L is positive; therefore N is so, and the principal element in qnestion consists 
wholly of positive terms. 
The above demonstration shows that the elimination of a from the %-system produces 
a new determinant equivalent to the original one, and in which the characters noted in 
the original one still remain. Should a occur in any other system or systems of elements 
of the new determinant beside the 1-system, it can, by repetitions of the same process, 
be eliminated thence. Ultimately, then, it will only remain in the 1-system, and there- 
fore only in the principal term of that system. Again, as it enters that term in the fii’st 
degree, it follows that the developed determinant will involve only the first power of a. 
Hence, as a may represent any of the quantities «, 5, c, ..., it is seen that no powers, but 
only products of these quantities, will appear in the developed determinant. 
Let us represent the determinant, after the elimination of a from all the elements but 
Ai, in the form 
A, 
B,2 . . . 
^21 
C 2 ... 
C2„ 
C„2 ... 
c„. 
Now let ali^ represent that term in Aj which involves a. 
minant which involves a will be 
ali^ 
a 
a. 
Then the portion of the deter- 
And here it is to be observed that a\ is positive, while the new determinant to which 
it is attached as a coefficient possesses all the characters of the old one. This determi- 
nant we can therefore transform in the same way, so as to eliminate any other letter b 
from all but a single principal element, which we shall suppose to contain it in a term 
5^2* That portion of the original determinant which involves ah will therefore assume 
the form 
dbW 
D, ... D, 
D„3 . . . D„. 
Ultimately, then, as the result of such processes continued, the portion of the original 
determinant which involves any particular combination of n letters selected from 
G . .. will consist of the product of a series of positive terms, each of which has 
appeared in some residual principal element. Every such combination being positive, 
it follows that the determinant itself consists solely of positive terms. ' 
