OF TESTIMONIES OR 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 a is present or not; then, considering p as a presumption for or against 
: 2 1 
z, according as p is greater or less than 5 
1. That presumption is strengthened if the events 2 and y are known to be 
jointly present, 7.¢., the probability of z is greater than p, if p is greater than 
2 but less than p in the contrary case. 
2. The strengthening of the presumption is greatest when cis least. In other 
words, the less likely the events 2 and y are to happen, the more does their actual 
concurrence strengthen the presumption, favourable or unfavourable, which either 
of them alone must afford. 
Sthly, If we suppose c and ¢ both to approximate to 0, the values of u and t 
also approximate to 0, and the ratio =e assumes at the limit the form = It 
may, however, be shown that its actual value at the limit is 
igs sea) cat ial ach intmnanicnial 
This is most readily obtained from (1) and (2), by rejecting the terms a’x2 and at, 
which we may do when wu and ¢ are infinitesimal. We thus find that w and ¢tend 
to assume the values ce’pg and ce'(1—p) (1—¢), whence 
as PY 
utt  — pq+(1—p) (1—g) 
It is interesting here to inquire whether the appearance of the limiting value 
=== is due merely to the smallness of cand ¢. 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 w 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 # and y, would receive strength, whatever the values of ¢ and ¢’ might be. 
And this expectation was verified. On taking for the data 
Prob. z=c, Prob. y=¢, Prob. zy=m, Prob. xz=cp, Prob. yz=ce'q 
and representing the sought value of Poe by w, I found, for the determina- 
nation of w, the equation 
(cp—mw) (¢q—mw) (1—w) =w(cl—p—ml—w) (cI—g—mi—w) . . (9) 
VOL. XXI. PART IV. 81 
