As a third example I shall choose the following problem: A number _u_ is taken at random, and _n_ is a given very large integer. What is the probable value of sin _nu_? This problem has no meaning by itself. To give it one a convention is needed. We _shall agree_ that the probability for the number _u_ to lie between _a_ and _a_+ is equal to [phi](_a_)_da_; that it is therefore proportional to the infinitely small interval _da_, and equal to this multiplied by _a_ function [phi](_a_) depending only on _a_. As for this function, I choose it arbitrarily, but I must assume it to be continuous. The value of sin _nu_ remaining the same when _u_ increases by 2[pi], I may without loss of generality assume that _u_ lies between 0 and 2[pi], and I shall thus be led to suppose that [phi](_a_) is a periodic function whose period is 2[pi]. The probable value sought is readily expressed by a simple integral, and it is easy to show that this integral is less than 2[pi]M_{_k_}/_n_^{_k_}, M_{_k_} being the maximum value of the _k_th derivative of [phi](_u_). We see then that if the _k_th derivative is finite, our probable value will tend toward 0 when _n_ increases indefinitely, and that more rapidly than 1/_n_^{_k_ - 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 [phi](_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?