A very simple derivation of this equation of Einstein has been given by Langevin of Paris[199] essentially as follows: From the kinetic theory of gases we have in which is the average of the squares of the velocities of the molecules, the number of molecules in a gram molecule, and the mass of each. Hence the mean kinetic energy of agitation of each molecule is given by . Since in observations on Brownian movements we record only motions along one axis, we shall divide the total energy of agitation into three parts, each part corresponding to motion along one of the three axes, and, placing the velocity along the -axis equal to , we have Every Brownian particle is then moving about, according to Einstein’s assumption, with a mean energy of motion along each axis equal to This motion is due to molecular bombardment, and in order to write an equation for the motion at any instant of a particle subjected to such forces we need only to know (1) the value of the -component of all the blows struck by the [Pg 269] molecules at that instant, and (2) the resistance offered by the medium to the motion of the particle through it. This last quantity we have set equal to and have found that in the case of the motion of oil droplets through a gas has the value We may then write the equation of motion of the particle at any instant under molecular bombardment in the form Since in the Brownian movements we are interested only in the absolute values of displacements without regard to their sign, it is desirable to change the form of this equation so as to involve and . This can be done by multiplying through by . We thus obtain, after substituting for its value , Langevin now considers the mean result arising from applying this equation at a given instant to a large number of different particles all just alike. Writing then for in which denotes the mean of all the large number of different values of , he gets after substituting for , and remembering that in [Pg 270] taking the mean, since the in the last term is as likely to be positive as negative and hence that , Separating the variables this becomes which yields upon integration between the limits and For any interval of time long enough to measure this takes the value of the first term. For when Brownian movements are at all observable, is or less, and since is roughly equal to we see that, taking the density of the particle equal to unity, Hence when is taken greater than about seconds, rapidly approaches zero, so that for any measurable time intervals or [Pg 271] and, letting represent the change in in the time