244 
PEOFESSOR BOOLE ON THE THEORY OF PROBABILITIES. 
the same hypothesis, when one or more of the variables w^, is wholly absent 
from that system. Thus if Xy were a common factor of all the terms of V, it would 
divide out from the numerators and denominators of the system, which would thus be- 
come a system of n simultaneous equations connecting the 7i—\ variables x^, x^ . . . x„. 
Considered with reference to these variables, therefore, the equations of the system would 
not be independent. 
All resulting inequations will be capable of expression under the one general form, 
4 - 052^^2 • • • 
the coefficients «i,« 2 , and h being positive, negative, or vanishing, numerical con- 
stants. For any inequation which presents itself in the form 
may be transformed into _ 
Again, the general inequation 
a:p,+a,p, . . . 5^0 
determines an inferior limit of when is positive, and a superior limit of p^ when a, 
is negative. 
For in the former case we have 
the second member of which is an inferior limit of ; and it will be observed that the 
calculated value of this member may be positive, as there is no general restriction on 
the signs of a^, . . . h. 
In the latter case, changing into — «i, and observing that is positive, we have 
the second member of which is a superior limit of^j. 
Lastly, the final system of inequations is totally independent of the numerical value 
of the coefficients of V. The only restriction is that these coefficients are positive. 
Proposition V. 
Let V he incomplete inform; then, provided that the eguations 
Vi V2_ v„_ 
Y — Pi? Y — Pi • • • Y — 
a-) 
a7'e independent with respect to the quantities x^, x^, • • • ^n? vend that the inequations of 
condition deducible hy the last proposition are satisfied, the equations will admit of one 
solution, and only one, in positive finite values of x^, x^, ... 
The proof of this proposition will, in its general character, resemble the proof of 
