30 SECTIONAL ADDRESSES 



system, and jnany are still trying to bring probability within the fold 

 of the old logic. I do not believe it can be done. This is not the 

 occasion, nor have I the capacity, for a deep argument on the place of 

 probability in logic, but one of the most convincing w^ays of seeing it 

 may be found in the consideration of another branch of physical theory, 

 the kinetic theory of gases. 



In the early days of kinetic theory the central problem was the law of 

 distribution of velocities of the molecules, and attempts were made to 

 prove the law absolutely from dynamics, but the process always failed. 

 Maxwell made the assumption that with the lapse of time a system of 

 molecules would pass through all possible phases. There are technical 

 difKculties in the discussion of this assumption which have never been 

 overcome, and it is quite uncertain if it is even true. Indeed Kelvin, 

 who disliked the whole kinetic theory, argued with some force that the 

 only examples anyone could give contradicted the principle — for example, 

 the motion of the planets. The greatest contribution to the subject 

 was that of Gibbs, who recognised that there had to be a big assumption 

 somewhere and made it quite frankly and without attempt at justification. 

 The works of Gibbs are not easy reading ; in both his great works he 

 attends to every detail with a particularity that is really rather tedious, 

 whereas his basic ideas are thrown at the reader almost without explana- 

 tion. The idea of a canonical ensemble is a really beautiful idea once you 

 understand it, but where does it come from ? An ensemble is an idea 

 which will be unfamiliar to many, so I had better explain it. We want 

 to know something about the behaviour of a complicated system com- 

 posed of a great many parts ; say we want to know the pressure of the 

 . gas in some vessel. If we tried to attack the question by pure mechanics, 

 we should be faced with an enormous number of mechanical equations 

 for the motions of the molecules, and even if these could be solved the 

 solution would be of no use, because it would depend on the initial 

 positions and velocities of the molecules, and these we should not know. 

 Instead of trying this impossible and useless task, Gibbs considers a very 

 large number of possible states of motion of the set of molecules, which 

 have some character in common such as their total energy, but which 

 are otherwise unrelated. Though each specimen of the motions is quite 

 independent of all the others, he looks at them all together ; this explains 

 the word ensemble — I do not know why he had to take a French word — • 

 and makes the assumption that the pressure of the gas is correctly given 

 by the average of all the specimens. The actual gas in the vessel at any 

 instant is one of the specimens ; in its motion it passes into configurations 

 corresponding ro others, but only after a fantastically long time would it 

 go through even a perceptible fraction of the whole ensemble. Gibbs 

 is assuming that the behaviour of the actual gas will be determined by 

 the average of the uncountable millions of specimens in the ensemble. 

 Almost at the start one finds oneself presented with the ensemble with 

 hardly an attempt to explain where it comes from or why it is right, 

 and the beginner is usually troubled by the fact that, though the subject 

 is obviously mechanical, all the mechanics he laboriously learnt in his 

 youth seem to have faded into comparative unimportance. There are 

 various kinds of ensemble, the chief of which is the canonical, correspond- 



