PEOFESSOE BOOLE ON THE THEOET OF PEOBABILITIES. 
233 
not appear in the final result, because the developed form of w may not contain any terms 
affected with the symbol When such terms do appear, the constant c admits of an 
interpretation indicating what new data are required to make the solution definite*. 
It is proper here to observe that the conditions of possible experience can be deter- 
mined as well jfiom the ‘ translated ’ as from the original form of the problem. That 
the results will agree is evident a priori, but it may be desirable to point out the 
analytical connexion of the two processes. I will take the example just considered, and 
then offer some general remarks on the subject. 
Eepresenting the events xy, yz, zx by s, f, v, the translated data would be found to be 
Prob. s=p, Prob. t=q, Prob. v=r, 
s, t, and V being connected by the explicit logical condition 
stv-\-stv ^=1. 
It is easily shown that the first member of this equation represents the sum of those 
combinations of the events s, t, v, with respect to happening or failing, which involve no 
logical contradiction. 
If, then, we represent under the above condition 
Prob. stv=X', Prob. stv=yJ, Prob. sfv=v', Prob. stv=.^\ 
we shall have 
V+v'=^, V+g'=r, 
V>0, ^^>0, *'^>-0, |^>0, 
-f- "b H” < 1 • 
This system of equations and inequations agrees with that employed in the previous 
solution, if we only make 
V=X, ^'=^, 
so that the ehmination of X', yJ, v\ ^ will lead to the same results as before. 
In general it may be observed that each combination of s, t, v which is possible with- 
out logical contradiction, gives, on substituting for their expressions in the simple 
terms of the original problem, either a single combination of those simple terms, or a 
sum of such combinations ; but the same combination of those simple terms will not arise 
from two different combinations of s, ^... It is clear from this that the systems of 
united equations and inequations arising in the two forms of the problem will be related 
in the following manner, viz. — 
For each positive quantity l! in the one set, there will exist either a single positive 
quantity X, or a sum of such quantities Xid-?t 2 +&c. in the other; but each such sum is 
inseparable, and the elements it is composed of are distinct from those of any other sum 
arising from any other of the quantities X'... It is evident, then, that the final results of 
elimination will be the same. The same formal processes which eliminate single quan- 
* Laws of Thought, p. 267. 
2 I 
MDCCCLXII. 
