What is at the instant _t_ the probable distribution of the minor planets? Or rather what is the probable value of the sine of the longitude at the instant _t_, that is to say of sin (_at_ + _b_)? We made at the outset an arbitrary convention, but if we adopt it, this probable value is entirely defined. Divide the plane into elements of surface. Consider the value of sin (_at_ + _b_) at the center of each of these elements; multiply this value by the surface of the element, and by the corresponding density of the imaginary matter. Take then the sum for all the elements of the plane. This sum, by definition, will be the probable mean value we seek, which will thus be expressed by a double integral. It may be thought at first that this mean value depends on the choice of the function which defines the density of the imaginary matter, and that, as this function [phi] is arbitrary, we can, according to the arbitrary choice which we make, obtain any mean value. This is not so. A simple calculation shows that our double integral decreases very rapidly when _t_ increases. Thus I could not quite tell what hypothesis to make as to the probability of this or that initial distribution; but whatever the hypothesis made, the result will be the same, and this gets me out of my difficulty. Whatever be the function [phi], the mean value tends toward zero as _t_ increases, and as the minor planets have certainly accomplished a very great number of revolutions, I may assert that this mean value is very small. I may choose [phi] as I wish, save always one restriction: this function must be continuous; and, in fact, from the point of view of subjective probability, the choice of a discontinuous function would have been unreasonable. For instance, what reason could I have for supposing that the initial longitude might be exactly 0°, but that it could not lie between 0° and 1°? But the difficulty reappears if we take the point of view of objective probability, if we pass from our imaginary distribution in which the fictitious matter was supposed continuous to the real distribution in which our representative points form, as it were, discrete atoms. The mean value of sin (_at_ + _b_) will be represented quite simply by (1/_n_){[Sigma] sin (_at_ + _b_)}, _n_ being the number of minor planets. In lieu of a double integral referring to a continuous function, we shall have a sum of discrete terms. And yet no one will seriously doubt that this mean value is practically very small. Our representative points being very close together, our discrete sum will in general differ very little from an integral. An integral is the limit toward which a sum of terms tends when the number of these terms is indefinitely increased. If the terms are very numerous, the sum will differ very little from its limit, that is to say from the integral, and what I said of this latter will still be true of the sum itself.