724 SCIENCE AND HYPOTHESIS 



a s.'mple integral, aod il is easy to show that this integral is smaller 



than 2 ^Mg UK being the maximum value of the nth derivative ot 



* K 

 <f> (u). We see then that if the K& derivative is finite, our mean 



value will tend towards zero when n increases indefinitely, and 



that more rapidly than ~^ . The mean value of sin nu when n is 



n 



very large is therefore zero. To define this value I required a con- 

 vention, 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 </> (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 role of what philosophers call the principle of sufficient reason, 

 and on the other hand of the importance of the fact that certain pro- 

 perties are common to all continuous functions. The study of prob- 

 ability 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 ignorance namely, those in which we know the law but 

 do not know the initial state of the system. I could multiply exam- 

 ples, but I shall take only one. What is the probable present distri- 

 bution of the minor planets on the zodiac ? We know they obey the 

 laws of Kepler. We may even, without changing the nature of the 

 problem, suppose that their orbits are circular and situated in the 

 same plane, a plane which we are given. On the other hand, we 

 know absolutely nothing about their initial distribution. However, 

 we do not hesitate to affirm, that this distribution is now nearly uni- 

 form. 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 time i.e., at the time 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 our minor planet 

 by a point in a plane namely, the point whose co-ordinates are 

 a and b. All these representative points will be contained in a cer- 

 tain region of the plane, but as they are very numerous this region 

 will appear dotted with points. We know nothing else about the dis- 

 tribution of the points. Now what do we do when we apply the 

 calculus of probabilities to such a question as this? 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 compelled to 

 make an arbitrary hypothesis. To explain the nature of this hypo- 

 thesis I may be allowed to use, instead of a mathematical formula, a 

 crude but concrete image. Let us suppose that over the surface of 

 our plane has been spread imaginary matter, the density of which is 



