PEOFESSOE BOOLE ON THE THEOEY OE PEOBABILITIES. 
243 
But we have already seen (Prop. II.) that the first member of this equation is essentially 
. V , . 
positive for positive values of Hence the function y varies by continuous 
increase, and on the hypothesis that the proposition to be proved is true for n—l 
variables, it is true for n variables. 
Therefore, connecting this with the former result, the proposition is true universally. 
Peoposition IV. 
If Y be an incomplete function^ some of the terms belonging to the compete form being 
wanting, but the terms 'present having their coefficients positive, it will in general be neces- 
sary not only that the guanUties p^, p^, . . -Pn should he positive fractions, but also that they 
should satisfy certain ineguations of the form 
ap,-\-a^p^...-\-a„p^-\-b>^,^ ^ 
in order that the system 
V, V, V„ , ■ ^ Vi 1 
y — III Y — l2-”Y — 
may admit of a solution in positive values of . x^. 
For let Ax^ x^Xt...he any term in V, A being a constant which is positive in all 
the terms, but which may be difierent in the different terms. Suppose that in there 
exist e terms like the above, and let the several ratios of these terms to V be denoted 
by X,, Xa ... > 1 ^. Then the ^th equation of the system (1.) will become 
Xj+Xa ... • • • • • • • • • • • (2-) 
and the system (1.) will be converted into a system of n equations of this nature. We 
will suppose that there exist m distinct quantities of the nature of Xi,X 3 ..i Xe in the first 
members of this transformed system, and we will represent these by Xj, Xa . . . X„j. Then, 
if these constitute all the ratios of the separate terms of V to V itself, we have a new 
equation, 
Xi+Xa ... +X„=1. . . , ... ... . . . . (3.) 
If they do not constitute all those separate ratios, we have, on the contrary, an inequa- 
tion, 
Xi+Xg... +X^<1 (4.) 
Lastly, the condition that Xj, Xg ... X„, are positive fractions, gives the inequations 
X,>0, X2>0 . . . X,„>0. .......... (5.) 
The conditions Xi<l, &c. are already implied in (3.) or (4.). 
The X quantities are thus subject to a system of united eguations and ineguations, from 
which they must be eliminated by the method already explained. 
The result of such elimination will be a final system of inequations connecting 
hi Pn • ‘ • Pn- Equations connecting these quantities can only present themselves when 
the equations of the original system are not independent, or, which really falls under 
2 e; 2 
