its limit, that is to say from the integral, and what I said of this latter will still be true of the sum itself.
Nevertheless, there are exceptions. If, for instance, for all the minor planets,
b = π/2 − at,
the longitude for all the planets at the time t would be π/2, and the mean value would evidently be equal to unity. For this to be the case, it would be necessary that at the epoch 0, the minor planets must have all been lying on a spiral of peculiar form, with its spires very close together. Every one will admit that such an initial distribution is extremely improbable (and, even supposing it realized, the distribution would not be uniform at the present time, for example, on January 1, 1913, but it would become so a few years later).
Why then do we think this initial distribution improbable? This must be explained, because if we had no reason for rejecting as improbable this absurd hypothesis everything would break down, and we could no longer make any affirmation about the probability of this or that present distribution.
Once more we shall invoke the principle of sufficient reason to which we must always recur. We might admit that at the beginning the planets were distributed almost in a straight line. We might admit that they were irregularly distributed. But it seems to us that there is no sufficient reason for the unknown cause that gave them birth to have acted along a curve so regular and yet so complicated, which would appear to have been expressly chosen so that the present distribution would not be uniform.
IV. Rouge et Noir.—The questions raised by games of chance, such as roulette, are, fundamentally, entirely analogous to those we have just treated. For example, a wheel is partitioned into a great number of equal subdivisions, alternately red and black. A needle is whirled with force, and after having made a great number of revolutions, it stops before one of these subdivisions. The probability that this division is red is evidently 1/2. The needle describes an angle θ, including several complete revolutions. I do not know what is the probability that the needle may be whirled with a force such that this angle should lie between θ and θ + dθ; but I can make a convention. I can suppose that this probability is ϕ(θ)dθ. As for the function ϕ(θ), I can choose it in an entirely arbitrary manner. There is nothing that can guide me in my choice, but I am naturally led to suppose this function continuous.
Let ε be the length (measured on the circumference of radius 1) of each red and black subdivision. We have to calculate the integral of ϕ(θ)dθ, extending it, on the one hand, to all the red divisions and, on the other hand, to all the black divisions, and to compare the results.
Consider an interval 2ε, comprising a red division and a black division which follows it. Let M and m be the greatest and least values of the function ϕ(θ) in this interval. The integral extended to the red divisions will be smaller than ΣMε; the integral extended to the black divisions will be greater than Σmε; the difference will therefore be less than Σ(M − m)ε. But, if the function θ is supposed continuous; if, besides, the interval ε is very small with respect to the total angle described by the needle, the difference M − m will be very small. The difference of the