OF TESTIMONIES OE JUDGMENTS. 637 



it is not known whether y is present or not, the probability of z is p, the same 

 probability being assigned to z, when it is known that y is present, but not known 

 whether x is present or not; then, considering p as a presumption for or against 



z, according as p is greater or less than ^. 



1. That presumption is strengthened if the events x and y are known to be 

 jointly present, i.e., the probability of z is greater than p, if p is greater than 



~, but less than p in the contrary case. 



2. The strengthening of the presumption is greatest when c is least. In other 

 words, the less likely the events x and y are to happen, the more does their actual 

 concurrence strengthen the presumption, favourable or unfavourable, which either 

 of them alone must afford. 



8thly, If we suppose c and d both to approximate to 0, the values of u and t 



u 



also approximate to 0, and the ratio — - assumes at the limit the form -x. It 



may, however, be shown that its actual value at the limit is 



pq 



(8) 



pq+(l-p)(l-q) 



This is most readily obtained from (1) and (2), by rejecting the terms a'u 2 and at 2 , 

 which we may do when u and t are infinitesimal. We thus find that u and t tend 

 to assume the values cc'pq and cc'(l-p) (1—q), whence 



u pq 



u + t "' pq+(l— p) (1 — q) 



It is interesting here to inquire whether the appearance of the limiting value 

 1 ~ TZT * s ^ ue mere ly to the smallness of c and c / . In studying this ques- 

 tion, it occurred to me that it is generally not the mere improbability of events, 

 or the mere unexpectedness of testimonies considered in themselves, but the im- 

 probability of the concurrence of such events or testimonies which gives to their 

 union the highest degree of force. I therefore anticipated, that, if I should in- 

 troduce among the primary data of the problem, the probability of the concur- 

 rence of the events x and y, assigning to it a value m, it would appear that, when- 

 ever m approached to 0, the presumptions with reference to the event z, founded 

 upon x and y, would receive strength, whatever the values of c and c might be. 

 And this expectation was verified. On taking for the data 



Prob. x=-c, Prob. y = c', Prob. ocy = m, Prob. xz — cp, Prob. yz = c'q 



and representing the sought value of p ' xyz by w, I found, for the determi- 

 nation of w, the equation 



(cp — mw) (c'q — mw) (1 — w)=w(cl— p — ml— w) (c'l— q — ml — w) . . (9) 

 VOL. XXI. PART IV. 8 I 



