234 
PEOPESSOE BOOLE ON THE THEOEY OE PEOB ABILITIES. 
titles in the one case, will eliminate the corresponding single quantities, or sums of single 
quantities, in the other. 
Simplification of the General liquations for the Solution of Questions in the 
Theory of Probabilities. 
Let us express the system (I.) in the form 
Y — Pi Y — T 
A + cC 
u — Y 5 
and let us suppose the quantities q... (and therefore 5 , L . . ) to be in number. Then 
ail the terms in V will be composed of products of s, t... s, t..., each term invohing 
either s or s, either t or t, &c., but not the combinations ss, tt, &c. Each term is there- 
fore homogeneous and of the ^^th degree. 
It follows, therefore, that if we divide the numerator and denominator of each of the 
fii’st members of the above system by s ^ i;.. and then make 
■X, 
— Xn 
— = a^3, &C., 
V 
and if at the same time we, for symmetry, change j?, q, r... into p2i • -Pm tke system 
^vill assume the following form. 
V, v„_ 
Y — V — • • • Y — P^’’ 
A + cC 
u — Y ’ 
in which V, A, C are formed from their former values by suppressing i, t, v, &c,, or, 
which is the same thing, changing each of them into unity, and then changing s, t, «... 
into Xi, Xo, X3..., while Vj consists of those terms of V which contain x^, Vg of those 
which contain x^, and so on. 
In its new form V is a rational and entire function of x^, x^, ...x^ not involving powers 
of those quantities, and with all its coefficients equal to unity. Again, as s, t, See. axe 
from the theory of their origin required to be positive proper fractions, ^1, x^, ... x„ are, 
from the natm’e of their connexion with s, t..., required to be positive quantities. Aud 
it is sufficient that Xi, x^, ... be determinable as positive quantities in order that s, t... 
may be determinable as positive fractions. 
Now we shall proceed to show that x^, x^, ... x^ are determinable as positive quantities 
precisely when ^1,^2, ...p„ satisfy the conditions of possible experience. We shall fm’ther 
show, as a consequence of this, that the value of the probability sought, when determined 
by the General Eule, will, under the same conditions, lie within such limits as if it were 
itself given by the same experience. In the order of this proof, we shall ffi’st demon- 
strate the theorems of pure Analysis upon which the conclusions depend, then in a 
distinct section make the particular application. 
