640 PROFESSOR BOOLE ON THE COMBINATION 



xyst + xys + xyt + xy + xs + x xyst + xys + xyt + xy + yt + y 

 ~~c~ c' 



_xyst + xys + xs xyst + xyt + yt 

 cp c'q 



xiist . , 



= — ,J — = xyst 4 xys + xyt + xy + xs + x + yt + y + l 



Equating the product of the third and fourth to that of the fifth and sixth mem- 

 bers of the above system, we have 



u = ce'pq = Prob. xywz 

 whence cc'(l-p) (1— p) = Prob. xywz 



And hence 



Prob. xi/wz pq 



• (15) 



Prob. xywz + Vrob. xywz PQ + i^—p) (1 — <?) 



Here it will be noted, that although the arbitrary constants c and c' were neces- 

 sarily introduced into the expression of the data of problem, they have no place in 

 its solution. The result, it will also be seen, agrees with (8) ; and it thus shows that 

 that formula would express the true solution of the problem originally proposed, 

 if it were permitted to neglect the circumstance that it is to the same fact that 

 the testimonies have reference, and so to regard their agreement as merely an 

 agreement in being true or in being false, but not in being true or in being false 

 about the same thing. 



*&■ 



Special Solution of Problem II. founder! upon the principle of the limit. 



36. In the present investigation we employ the principle stated in Art. 24, 

 our object being to determine the mean between p and q, when they represent 

 probabilities founded upon different judgments, just as in Art. 25 we have deter- 

 mined the mean between p and q, when they represent different observed values 

 of a physical magnitude. 



To the previous data, viz., 



Prob. x=c, Prob. y — c', Prob. xz=cp, Prob. yz=&q . . (1) 



we now add, as the supposed « priori value of Prob. z, 



Prob. z = r . (2) 



From these collective data we determine the fraction 



Prob. xyz Prob. xyz 



Prob. xy 0r p ro b. xyz + Prob. xy ~z 



representing the a posteriori value of Prob. z, and, equating the a priori and a 

 posteriori values, determine x. The principle upon which the investigation pro- 

 ceeds, is, that we attribute to the mean strength of the probabilities p and q such 

 a value, that if the mind had previously to the evidence been in the state of ex- 

 pectation which that value is supposed to measure, the evidence would not have 



