The first procedure was to find how badly Stokes’s Law failed in the case of our drops. This was done by plotting the apparent value of the electron against the observed speed under gravity. This gave the curve shown in Fig. 4, which shows that though for very small speeds varies rapidly with the change in speed, for speeds larger than that corresponding to the abscissa marked 1,000 there is but a slight dependence of on speed. This abscissa corresponds to a speed of .1 cm. per second. We may then conclude that for drops which are large enough to fall at a rate of 1 cm. in ten [Pg 99] seconds or faster, Stokes’s Law needs but a small correction, because of the inhomogeneity of the air. Fig. 4 To find an exact expression for this correction we may proceed as follows: The average distance which a gas molecule goes between two collisions with its neighbors, a quantity well known and measured with some approach to precision in physics and called “the mean free path” [Pg 100] of a gas molecule, is obviously a measure of the size of the holes in a gaseous medium. When Stokes’s Law begins to fail as the size of the drops diminish, it must be because the medium ceases to be homogeneous, as looked at from the standpoint of the drop, and this means simply that the radius of the drop has begun to be comparable with the mean size of the holes—a quantity which we have decided to take as measured by the mean free path . The increase in the speed of fall over that given by Stokes’s Law, when this point is reached, must then be some function of . In other words, the correct expression for the speed of a drop falling through a gas, instead of being as Arnold showed that it was when the holes were negligibly small—as the latter are when the drop falls through a liquid—should be of the form If we were in complete ignorance of the form of the function we could still express it in terms of a series of undetermined constants , etc., thus and so long as the departures from Stokes’s Law were small as Fig. 4 showed them to be for most of our drops, we could neglect the [Pg 101] second-order terms in and have therefore Using this corrected form of Stokes’s Law to combine with (9) (p. 20), we should obviously get the charge in just the form in which it is given in (13), save that wherever a velocity appears in (13) we should now have to insert in place of this velocity . And since the velocity of the drop appears in the ³⁄₂ power in (13), if we denote now by e the absolute value of the electron and by as heretofore, the apparent value obtained from the assumption of Stokes’s Law, that is, from the use of (13), we obtain at once In this equation can always be obtained from (13), while is a known constant, but , , and are all unknown. If can be found our observations permit at once of the determination of both and , as will be shown in detail under Section VI (see p. 105).