Page:Popular Science Monthly Volume 80.djvu/441

for, while seeing the oil drops dance may satisfy the average man, it will not satisfy the scientist, for he is never content until he has two parallel columns headed, respectively, "calculated" and "observed" values. How shall we set about obtaining such parallel columns? The way was blazed by Einstein in 1905. He showed that if a body like one of our minute oil drops is dancing about in a resisting medium subjected to no forces but those arising from its own energy of agitation, that is, from the bombardment of the surrounding molecules, the mean distance which it will drift in a given time, say ten seconds, from its position at the beginning of this time, can be computed in terms of three factors: (1) its energy of agitation, (3) a resistance factor of the medium, and (3) the length of the time interval through which the drift is observed.^[1] But this same quantity can also be easily and directly observed in our experiment by simply balancing the force of gravity upon the drop by the force of an electrical field in the manner already described, and then noting over how large a distance on the average it wiggles in a given time by virtue of its energy of agitation. In the actual experiments we took, in the case of each drop, the mean of several hundred observations on the distance moved in ten seconds in a vertical direction over a set of horizontal scale divisions placed in the eye-piece of the observing telescope; for Einstein's theory was developed in such a way that the movements to right and left did not need to be considered. The computed and the observed values of this average displacement were in every case in so perfect agreement as to satisfy the most skeptical of scientists that the kinetic theory can successfully meet a rigorous and exacting kind of quantitative test.

But in order to show how free from uncertainties of any sort are the results of this comparison it will be necessary to say just a word more about the theory, for the question is at once raised "how, in computing the theoretical value of the average displacement of the drop, do you obtain the first two of the factors in terms of which this displacement is given, namely, the kinetic energy of agitation of the drop and the resistance factor of the medium?" We obtain a partial answer to this question when we remember that one of the fundamental assumptions of the kinetic theory is that the energy of agitation of a molecule is determined by temperature alone, and is independent of

1. Einstein's actual equation is $\scriptstyle D^{2}=4/3\ .\ E/K\ .\ t$, in which $D^{2}$ is a quantity obtained by squaring each individual displacement and then taking the mean of these squares, E is the mean kinetic energy of agitation of the drop, K is a resistance factor depending upon both the medium and the drop, and t is the length of the time interval used. If the average displacement D is used instead of the average square of the displacements $D^{2}$ the correct form of the equation is

   $\scriptstyle D={\sqrt {\frac {8}{3\pi }}}{\frac {E}{K}}t$.