240 
PEOFESSOE BOOLE ON THE THEOEY OP PEOBABILITIES. 
Therefore by the last proposition the developed determinant will consist of products 
(without powers higher than the first) of different terms of V, and the coefficients of all 
such products will be positive. 
Therefore the determinant will be expressible as a rational entire function of X 2 , x„ 
with positive coefficients. 
The rapidity with which the complexity of the determinant increases as the number 
of variables increases is remarkable. For example, if n—2 and N —axy-\-hx-\-cy-\-d, 
the determinant is 
axy-\-hx-\-cy-{-d axy-\-hx axy-\-cy 
axy-\-bx axy-{-bx axy 
axy-\-cy axy axy-{-cy\ 
and its calculated value will be found to be 
abcx-y'^ + abdx^y-\-acdxy^ + bcdxy, 
consisting of four positive terms. 
But if and 
V = axyz + byz + cxz + dxy + ex -^-fy-^rgz + K 
the developed determinant will consist of fifty-eight positive terms. Its calculated value 
will be found in the Memoir on Testimonies and Judgments. 
Proposition III. 
The functions V, V„ Vj, . . . V„ being defined as above, ifYbe com/plete inform, i. e. if 
it consist of all the terms which according to definition it can contain, each with a positive 
coefficient, then the system of eguations 
V,_ V,_ y^_ . 
will, whenp^,p 2 -> • • -Pn are proper fractions, admit of one solution, and only one, inpositive 
values of x^, x^,... x^. 
We shall show, first, that the above proposition is true when n=\, secondly, that on 
the hypothesis that it is true for n—1 variables, it is true for n variables. Hence it will 
follow that it is true generally. 
Suppose w=l. Then V =aXi-\-b, whence the system (1.) reduces to the single equation 
axi + b 
1 c(l-Pi) 
whence, since a and b are positive, and ^ is a positive fraction, x^ is positive. 
Thus the proposition is true when w=l. 
Now, let 0^1=0, and let x^, x^. . .x^^tQ determined to satisfy the last n—\ equations 
of the system (I.). These n—1 equations will, when a^i=0, form a system of the same 
