60 The Law of Distribution [OH. iv 



decrease as we recede from this point in either direction along the trajectory 

 through the point. In other words, it is infinitely probable that the value 

 H = H! is a maximum value of H for the trajectory through the point. 



70. It may, perhaps, still be thought paradoxical that dH/dt is not zero 

 at each of these maxima. The explanation is that the variation of H is not 

 governed by the laws of the differential calculus, since this variation is not, 

 strictly speaking, continuous. The value of H is constant between collisions 

 of the molecules, and changes abruptly at every collision. When the number 

 of molecules in the gas is infinite, the interval between successive collisions 

 will become infinitely small, but in general the variation in H will not be 

 continuous. For obviously the differential coefficients of H vanish between 

 collisions and become infinite at every collision, so that H, regarded as a 

 function of t as we follow any trajectory, will be a function of the well-known 

 type which possesses an infinite number of maxima and minima within a 

 finite range of the variable. We can, however, "smooth out" the curve 

 obtained for H as a function of t and in this way obtain the function H 

 as a continuous function of the time. This is the H contemplated by the 

 analysis of Chapter II. But now we can also see that there is no reason 

 to suppose that dH/dt will vanish when H attains a maximum value, but 

 that on the contrary H will in general change sign abruptly at such a point. 

 It is therefore clear that, averaged over all systems which have a given f, 

 dH/dt will be negative except when f is the law of distribution for the normal 

 state, in which case it is zero. This result is now in agreement with that of 

 Chapter II. 



