The probable value sought is readily expressed by a simple integral, and it is easy to show that this integral is less than
2πMk ⁄ nk,
Mk being the maximum value of the kth derivative of ϕ(u). We see then that if the kth derivative is finite, our probable value will tend toward 0 when n increases indefinitely, and that more rapidly than 1/nk−1.
The probable value of sin nu when n is very large is therefore naught. To define this value I required a convention; but the result remains the same whatever that convention may be. I have imposed upon myself only slight restrictions in assuming that the function ϕ(a) is continuous and periodic, and these hypotheses are so natural that we may ask ourselves how they can be escaped.
Examination of the three preceding examples, so different in all respects, has already given us a glimpse, on the one hand, of the rôle of what philosophers call the principle of sufficient reason, and, on the other hand, of the importance of the fact that certain properties are common to all continuous functions. The study of probability in the physical sciences will lead us to the same result.
III. Probability in the Physical Sciences.—We come now to the problems connected with what I have called the second degree of ignorance, those, namely, in which we know the law, but do not know the initial state of the system. I could multiply examples, but will take only one. What is the probable present distribution of the minor planets on the zodiac?
We know they obey the laws of Kepler. We may even, without at all changing the nature of the problem, suppose that their orbits are all circular, and situated in the same plane, and that we know this plane. On the other hand, we are in absolute ignorance as to what was their initial distribution. However, we do not hesitate to affirm that their distribution is now nearly uniform. Why?
Let b be the longitude of a minor planet in the initial epoch, that is to say, the epoch zero. Let a be its mean motion. Its longitude at the present epoch, that is to say at the epoch t, will be at + b. To say that the present distribution is uniform is to say that the mean value of the sines and cosines of multiples of at + b is zero. Why do we assert this?
Let us represent each minor planet by a point in a plane, to wit, by a point whose coordinates are precisely a and b. All these representative points will be contained in a certain region of the plane, but as they are very numerous this region will appear dotted with points. We know nothing else about the distribution of these points.
What do we do when we wish to apply the calculus of probabilities to such a question? What is the probability that one or more representative points may be found in a certain portion of the plane? In our ignorance, we are reduced to making an arbitrary hypothesis. To explain the nature of this hypothesis, allow me to use, in lieu of a mathematical formula, a crude but concrete image. Let us suppose that over the surface of our plane has been spread an imaginary substance, whose density is variable, but varies continuously. We shall then agree to say that the probable number of representative points to be found on a portion of the plane is proportional to the quantity of fictitious matter found there. If we have then two regions of the plane of the same extent, the probabilities that a representative point of one of our minor planets is found in one or the other of these regions will be to one another as the mean densities of the fictitious matter in the one and the other region.