PEOFESSOE BOOLE OK THE THEOET OF PEOBABILITIES. 
245 
Proposition III. It will be shown that when we assign to x any value between the 
limits 0 and infinity, the quantities x^ ... x^ will admit of determination from the 
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last n—\ equations of the system as positive finite quantities, and the function ^ will 
receive a value falling within the limits assigned by Proposition IV. to the quantity ; 
that when x^ is equal to 0 or infinity, x^^ X 3 ... x^ will admit of determination either 
as positive finite quantities, or as limits (0 and 00) of such quantities ; and that these 
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values together will give to ^ a value coinciding with the highest of the inferior, or 
lowest of the superior limits of _pi, as determined by Proposition IV. ; that when x^ 
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varies from 0 to co, x^, X 3 ... x„ being determined as above, ^ will vary by continuous 
increase from the highest of the inferior to the lowest of the superior limits of 
and once in its variation become equal to Thus the truth of the proposition for 
n variables will flow necessarily from its assumed truth for n — 1 variables. And on this 
ground it will be shown that it may ultimately be reduced to a direct dependence upon 
Proposition III. 
In the system (I.) let x^ receive any finite positive value, and let V by the substitu- 
tion of this value become U ; the last n—\ equations of (I.) will thus assume the form 
Ug Ug Un /f) \ 
XJ — XJ — Pa • • • XJ —Pni (2-) 
in which the quantities ^2, ^>3 . . .^„ satisfy the conditions to which the direct application 
of Proposition IV. to this reduced system of equations would lead. 
For what is important to notice in the change from V to U is, that any two terms in 
V which differ only in that one contains Xi and the other does not, reduce to a single 
term in U. The effect of the change upon the primary system of equations and inequa- 
tions formed in the application of Proposition IV. to the system (I.) is the following: — 
1st. The equation between Xj, Xj . . . derived from the first equation of (1.) will be 
annulled. 
2ndly. In the remaining equations connecting Xj, X2 . . . some^a^V5 of those quantities 
may be replaced by single quantities, with corresponding changes in the inequations. 
Thus if Xi + Xj be replaced by (m, the inequations 
Xi>.0, X2>-0 
will be replaced by what they before implied., viz. 
iW<>0. 
But these changes do not affect the truth of the relations, or introduce any new rela- 
tions. They cannot, therefore, lead to any new final conditions. The conditions con- 
necting _p2? i>3 • • • Pn-> in accordance with Proposition IV. in the system (2.), must have 
been already involved in the equations connecting ^1, ^2 . . .^„ in the system (1.). 
Hence by hypothesis the system (2.) gives one set of positive finite values of 
