u lies between 0 and 2π, and I shall thus be led to suppose that Φ(a) is a periodic function whose period is 2π. The mean value that we seek is readily expressed by a simple integral, and it is easy to show that this integral is smaller than 2πMKnᴷ, MK being the maximum value of the Kth derivative of Φ(u). We see then that if the Kth derivative is finite, our mean value will tend towards zero when n increases indefinitely, and that more rapidly than 1nᴷ⁻¹. The mean value of sin nu when n is very large is therefore zero. To define this value I required a convention, but the result remains the same whatever that convention may be. I have imposed upon myself but slight restrictions when I assumed 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 now come to the problems which are connected with what I have called the second degree of
