606 PROFESSOR BOOLE ON THE COMBINATION 
Determination of the Conditions of Possible Haperience, 
13. To explain the method of effecting this object, by an example, I will first 
symbolically express the problem of Art. 11. 
Let us then represent rain by 2, snow by y, and wind by z. The problem in 
question then takes the following form:— 
Given Prob. zy=p, Prob. yz=g, Prob. az=r ile May 
Required Prob. xyz 5 : 5 5 : ‘ (2) 
The value required we shall represent by w. It is our present object, not to solve 
this problem, but to ascertain the conditions which must connect p, g, and 7, 
in order that the data may be possible, with the corresponding limitations of w. 
For if w were itself determined by experience, it would be subject to conditions 
of possibility similar to those which govern p, g, and *. 
Now let us write, resolving the events in the problem into the possible alter- 
nations out of which they are formed, 
Prob. ayz=u, Prob. ayz=A, _—~Prob. wzy=—, Prob. yza=¥. 
We have then 
utA=p, Uut+Vv=q, ut p=r . ; : (8) 
The first of these equations only expresses that the probability of the concurrence 
of « and y is equal to the probability of the concurrence of a, y, and z, and the 
probability of the concurrence of # and y without z. To the equations (3) we 
must now add the inequations 
a= 0; K=O; ps0,  vS0; 4 
utAt+ptvel Deepal Sy 
expressing the conditions to which wu, A, 4, y, 1st, as probabilities, and, 2dly, as 
probabilities which do not altogether make up certainty, are subject. 
First, we will eliminate A, », and y. Their values found from (3) are 
A=p—uU p=r—u y=q—-U. 
Substituting these in (4) we have 
us0 p—usd q-us9 r—us0 
ptqtr—2ue_l, 
Whence, 
UEP, UV=q WET, | 
ati u= AP ecaee. f : : c (5) 
Such are the conditions to which the quantity w is subject, conditions which the 
value of Prob. ayz must @ priori satisfy. 
To determine the conditions connecting p, g, and 7, we must from (5) eliminate 
u. Now, if we have any two inequations of the form 
U=a usb 
