but the first terms may be readily found by actual reversion of (3),

were published in the Berlin Astronomisches Jahrbuch for 1834, and are reprinted* at the end of my article on the Theory of Probabilities, in

* The increasing importance of this Table makes it worth while here to state that it will shortly appear in a comparatively popular Essay on Probabilities on which I am now engaged, and also, as I am informed, in the Article on the subject in the new edition of the Encyclopaedia Britannica.

the Encyclopaedia Metropolitana. By means of this table, and its differences, and the formulæ

Thus, u, Au, &c. being taken positively as in the table (remembering that A'u is really negative, and that Ax = '01), we have

In the case before us we have, writing for X,

nearly.

+

4√2n (1 − 2 w) √π (1 − ☎) (€ ̄μ2 — € ̄”') + (13π2 − 13π + 1) (μ€ ̄μ2 — v ε−12)

Let a = w and 1: then

0, b = w + 0, where 0 is a small fraction both of w

and the presumption that p, the probability of the arrival of A, lies between

In the formula of Laplace and M. Poisson, the result has

The latter is of a lower order than the former, and may safely be rejected. By taking successively for the limits

we find that the presumption of p lying between the several pairs of
limits stated in the first column is expressed by the formulæ in the second.

the first and third become equal when ☎ = ; that is, when A happens as often as B in n trials: a result which we might have looked for à priori. It also appears that when is less than, it is more likely that Ρ should exceed than fall short of it; which is in accordance with another result of the theory, namely, that the chance of drawing A at the (n + 1)th trial is

v + 1 , v + w. 2

v

which is nearer to than

v + w

Let w = 1 − × where « is small (not being less than).

= λκ

Let the limits be a 1-λ and b = 1; where x is greater than unity, or μ is negative,

and is positive and infinite. We have also

so that if happen n (1 - x) times out of n, the presumption that its A probability lies between 1-AK, and 1 is

1), and

Next, let = —K, where x is small (not being less than ),

K

let a = 0, b = 1; that is, required the presumption that the less frequent (slightly) of two events is the less probable. Then μ is infinite and negative, and is positive and derived from

v2 = n(1 − k) log (1 − 2) + n ( + x) log (1 + 2k)