644 PROFESSOR BOOLE ON THE COMBINATION 
the values which they respectively give to the probability of the event 2 are 7, 
DP, . - Px» The result is 
1 
(pps ? “Pai 
(ae - + Pn ): 4 (d-p) (l—p,) . . d—p,) \e IS 
r= 
This is the general formula of the mean in reference to judgments, and much as 
it differs from the formula of the mean, in reference to the observations of a 
physical magnitude, some remarkable points of analogy exist. I will notice but 
one. The arithmetical mean is not altered if to the quantities among which it is 
taken we add another equal to the previous mean. Thus we have 
Pit Po ++ +Pn+1 — Pit Pa» - + Pn 
n+l n 
provided that +1 aftr Or representing td. +2 by P, we have 
de +1 =e n 
provided that 
We egal . 5 5 a (2) 
The same relation may readily be shown to hold also, if P, represent the mean of 
judgment, as expressed in (1). 
39. The following is a brief summary of the conclusions established in this 
paper. 
1st, The solution of the problem of astronomical observations by the logical 
theory of probabilities is, in its general form, indefinite. 
2ndly, It becomes definite, if we introduce the general principle of means. The 
result is in accordance with the usual formule, but expresses the so-called weights 
of the observations as determinate functions of certain probabilities relating to 
the correctness of the observations, and the character of the observers. 
3dly, When, as respects the two last elements, the observations are considered 
equal, the formula is reduced to the expression of the arithmetical mean. 
4thly, The complete solution of the problem of the combination of two proba- 
bilities of an event founded upon different testimonies or judgments is indefinite, 
but admits, in various cases, of being reduced to a definite form. 
5thly, This indefiniteness is due to the circumstance indicated by the formula, 
that the strength of the probabilities in combination is due, not to the strength 
of the separate probabilities alone, but also to the degree of unexpectedness of 
the testimonies or judgments themselves. 
6th/y, Combined presumptions, whether for or against an event, are generally 
strengthened by the unexpectedness of the combination. 
Tihly, When probabilities as p,,p,,. .p, are in a high degree cumulative, owing 
