250 
PEOFESSOE BOOLE ON THE THEOET OE PEOBABILITEES. 
reducing transformations are all definite, the above indefinite forms cannot present 
themselves in the last stage of the problem when the original equations are independent 
and admit of definite solution. 
The proposition is therefore established. 
Application. 
The general system of algebraic equations upon which the solution of questions in 
the theory of probabilities depends, is a particular case of that discussed in Propo- 
sition V. Its peculiarity is, that all the coefiicients which appear in the function V are 
equal to unity. 
The conditions of possible experience, as determined by the method, agree with the 
conditions shown in Proposition IV. to be necessary, and in Proposition V. to be suffi- 
cient, in order that x^ • . . may be determinable as positive finite quantities. For 
in both cases the quantities A,, Ag, &c. correspond to the different terms in V, and in 
both cases the equations among those quantities depend simply on the forms of the 
functions Vj, Vg . . . V„, and therefore ultimately on the form of V, irrespectively of the 
values of the positive coefficients of V. In both cases the systems of inequations are 
the same. 
It follows, therefore, that precisely when the data represent a possible experience, the 
probabilities of the ideal events from which in the process of solution the problem is 
mentally constructed admit of determination as positive proper fractions. 
Again, as the process for determining the a priori limits of the probability sought 
rests ultimately upon the assumption that the ratio of any term or partial aggregate of 
terms in V to V itself is a positive fraction, and as this assumption is satisfied when 
x^, X2. . .x„ are positive quantities, it follows that the calculated value of the probability 
sought will always lie within the limits which it would have had if determined by 
observation from the same experience as the data. 
But though the test last mentioned is one which must necessarily be satisfied by a 
true method, it is of infinitely less theoretical importance than that from which it is 
derived, viz. the test which consists in the absolute connexion between possibility in 
the data and formal consistency in the method. 
As the conclusions of Propositions IV. and V. depend upon the form of the function V 
and the fact that its coefficients are positive, it follows that if in the application of the 
method to questions of probability we substituted any other positive values for unity in 
the coefficients of V, leaving the rest of the process as before, we should still be able to 
determine x^^ x^, ...x„ as positive quantities, or as limits of such, and the altered value oi 
the probability sought would still be consistent with the experience from which the data 
are supposed to be derived. It would, however, properly speaking, be a value of inter- 
polation, not a probability. 
I will close with a few remarks upon the general nature of the method, and of the 
solutions to which it leads. 
