In equation (28) let be the time required by the particle, if there were no Brownian movements, to fall between a series of equally spaced cross-hairs whose distance apart is . In view of such movements the particle will have moved up or down a distance in the time . Let us suppose this distance to be up. Then the actual time of fall will be , in which is now the time it takes the particle to fall the distance . If now is small in comparison with , that is, if is small in comparison with (say ⅒ or less), then we shall introduce a negligible error (of the order ¹⁄₁₀₀ at the most) if we assume that in which is the mean velocity under gravity. Replacing then in (28) (, by in which is the square of the average difference between an observed time of fall and the mean time of fall [Pg 154] , that is, the square of the average fluctuation in the time of fall through the distance , we obtain after replacing the ideal time by the mean time In any actual work will be kept considerably less than ⅒ the mean time if the irregularities due to the observer’s errors are not to mask the irregularities due to the Brownian movements, so that (29) is sufficient for practically all working conditions.[88] The work of Mr. Fletcher and of the author was done by both of the methods represented in equations (28) and (29). The 9 drops reported upon in Mr. Fletcher’s paper in 1911[89] yielded the results shown below in which is the number of displacements used in each case in determining or . TABLE XIV 1.68 125 1.67 136 1.645 321 1.695 202 1.73 171 1.65 200 1.66 84 1.785 411 1.65 85 When weights are assigned proportional to the number of observations taken, as shown in the last column of Table XIV, there results [Pg 155] for the weighted mean value which represents an average of 1,735 displacements, or , as against , the value found in electrolysis. The agreement between theory and experiment is then in this case about as good as one-half of 1 per cent, which is well within the limits of observational error. This work seemed to demonstrate, with considerably greater precision than had been attained in earlier Brownian-movement work and with a minimum of assumptions, the correctness of the Einstein equation, which is in essence merely the assumption that a particle in a gas, no matter how big or how little it is or out of what it is made, is moving about with a mean translatory kinetic energy which is a universal constant dependent only on temperature. To show how well this conclusion has been established I shall refer briefly to a few later researches.