412 THE SCIENTIFIC MONTHLY 



The principle of Least Action, enunciated by Maupertius in 1744 

 but first put into correct mathematical form by W. R. Hamilton a 

 century later, is essentially one which demands mathematical treatment. 

 The Action of a system is a certain function depending on the velocities 

 and relations of its parts which takes a minimum value whenever the 

 system moves under natural laws. The process of discovery of this 

 minimum leads to the differential equations of motion of the system 

 and thus includes a complete statement of the problem. Since the 

 initial form of the function is an integral, its mathematical treatment 

 consists in finding the least value of this integral and thus becomes a 

 problem in the calculus of variations to which considerable attention 

 has been given by pure mathematicians, especially during the last two 

 or three decades. While the physical consequences of the principle 

 have been less developed than those of energy, there appears to be a 

 growing feeling as to its fundamental importance and the aid of the 

 mathematician in solving the problems which it raises, will become 

 increasingly necessary. 



The study of the properties of a system containing a large number 

 of particles not fixed relatively to one another, now generally studied 

 under the term, statistical mechanics, has penetrated into several 

 branches. It is to be understood here that the question is not one of 

 finding the separate motions of the various particles but to try and 

 find out such properties of the system as can be deduced from averages. 

 It is probable that as long as we cannot observe the motions of the 

 separate particles, we should be able to deduce in this way most if not 

 all the properties of the system that we are able to observe. Maxwell 

 and Boltzmann founded the subject from this point of view, applying 

 it to the kinetic theory of gases, while J. W. Gibbs was largely respon- 

 sible for its application to thermodynamics. In astronomy the present 

 century has seen it applied to the motions and positions of the stars, 

 thus opening the way to a knowledge of the outlines of the construc- 

 tion of the stellar universe. Mathematically these questions are 

 obviously very similar to those parts of probability which deal with 

 errors of observation and thus form a continuation of the development 

 of that subject. 



The mathematics of continuous media has received very complete 

 development during the century and, besides the earlier investigations 

 into the motions of fluids and elastic solids, has been applied to the 

 so-called luminiferous ether and finally by Maxwell to the whole elec- 

 tric field. In all this work the continuity of the medium is a funda- 

 mental axiom involving the hypothesis that no action can take place 

 without its presence. Further, the Newtonian laws of motion were 

 assumed as fundamental and. time and length were regarded as un- 

 changeable separate entities. The classic experiment of Michelson and 



