I may choose ϕ 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) Σ 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.
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).