PEOFESSOE BOOLE ON THE THEOET OE PEOBABILITIES. 
237 
Thus a has been eliminated from the %-system, and as the process has not affected any 
elements but those which belong to the w-system, it will not affect the relations under 
which a enters into the other systems. 
Consider then any other quantity h in the set «, e, then by hypothesis the coeffi- 
cients of h in any line or column of elements 
may be represented by 
.A-ii, A.;», . . or A-ij, Ao;, . . 
/3i, jSg, . . (3„ being an arbitrary set of quantities which are the same for all lines or columns, 
while (Jiji differs for different lines or columns, and vanishes for those in which b does not 
enter. 
It is to be noted that as A,y=Aji, we have in general 
while as the principal elements of the determinant (I.) are positive, we have always 
fijl3i=a positive quantity. 
Now reverting to the derived determinant (B.), we see that its ^th line or column of 
elements will be 
Aji, A, '2, . . . A,-„ ^Aji, 
and its^'th line or column 
Aji, Aj2'i • • • Aj’^"~ 
supposing ^ and j to be both less than n. 
In these lines or columns the successive coefficients of h will therefore be 
^,.f32, . . . 
hA’ • • • 
which stand to each other in the constant ratio 
Now let_y=w. The coefficients of h in the wth line or column of (B.) are obviously 
f^nA—'^i^iA, ••• f^nA—^^y'iA+^AA, 
of which the last term may be reduced as follows, 
so that the series of coefficients of b becomes 
^f^l)(A'“^(^>)’ 
and they are now seen to stand to those of b in the ^-line and column in the constant 
ratio X/A, : 
We have, lastly, to prove that the new principal element A„ — 2XAi„-l-X^Ai is positive. 
Let N be the coefficient of any one of the quantities «, c . . . in the above element, 
L its coefficient in the principal element A^, and M its coefficient in each of the sub- 
ordinate elements common to the two systems of which the above are the respective 
