 On the elementary electrical charge and the Avogadro constant, physical review volume 2, number 2, by Robert Andrews Millican. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Sections 1 through 4 Section 1, Introductory The experiments herewith reported were undertaken with the view of introducing certain improvements into the oil drop method of determining E and capital N, and thus obtaining a higher accuracy than had before been possible in the evaluation of these most fundamental constants. In the original observations by this method, such excellent agreement was found between the values of E derived from different measurements, 1c, page 384, that it was evident that if appreciable errors existed, they must be looked for in the constant factors entering into the final formula, rather than in inaccuracies in the readings or irregularities in the behavior of the drops. Accordingly, a systematic redetermination of all these constants was begun some three years ago. The relative importance of the various factors may be seen from the following review. As is now well known, the oil drop method rested originally upon the assumption of Stokes' law, and gave the charge E on a given drop through the equation. Equation 1, E sub N, equals four-thirds pi, times open parenthesis, nine eta over two, close parenthesis to the three-halves power, times open parenthesis fraction, numerator one, denominator g, times open parenthesis, sigma minus rho, close parenthesis, and fraction, close parenthesis, to the one-half power, times fraction, numerator, open parenthesis, v sub one, plus v sub two, close parenthesis, times v sub one to the one-half power, denominator, capital F, and fraction, in which eta is the coefficient of viscosity of air, sigma, the density of the oil, rho, that of the air, v sub one, the speed of descent of the drop under gravity, and v sub two, its speed of ascent under the influence of an electric field of strength, capital F. The essential feature of the method consisted in repeatedly changing the charge on a given drop by the capture of ions from the air, and in thus obtaining a series of charges with each drop. These charges show the very exact multiple relationship under all circumstances, a fact which demonstrated very directly the atomic structure of the electric charge. If Stokes' law were correct, the greatest common divisor of this series of charges should have been the absolute value of the elementary electrical charge. But the fact that this greatest common divisor failed to come out a constant when drops of different sizes were used, showed that Stokes' law breaks down when the diameter of a drop begins to approach the order of magnitude of the mean-free path of a gas molecule. Consequently, the following corrected form of Stokes' law for the speed of a drop falling under gravity was suggested. Equation two. V sub one equals two-ninths times fraction numerator. G a squared times open parenthesis sigma minus rho close parenthesis. Denominator eta. End fraction. Open brace one plus capital A L over A. Close brace. In which A is the radius of the drop, L the mean-free path of a gas molecule, and capital A an undetermined constant. It is to be particularly emphasized that the term in the brackets was expressly set up merely as a first-order correction term in L over A, and involves no theoretical assumptions of any sort. Further, that the constant capital A was empirically determined through the use only of small values of L over A, and that the values of E and capital N obtained were therefore precisely as trustworthy as were the observations themselves. This fact has been repeatedly overlooked in criticisms of the results of the oil drop method. Calling then E sub one, the greatest common divisor of all the various values of E sub N, found in a series of observations on a given drop, they resulted from the combination of one and two, the equation, equation three, E open parenthesis one plus capital A L over A, close parenthesis to the three halves power, equals E sub one. Or equation four, E equals fraction numerator E sub one, denominator open parenthesis one plus capital A L over A, close parenthesis to the three halves power, and fraction. It was from this equation that E was obtained after capital A had been found by a graphical method, which will be more fully considered presently. The factors then which enter into the determination of E are, one, the density factor sigma minus rho, two, the electric field strength capital F, three, the viscosity of air eta, four, the speeds V sub one and V sub two, five, the drop radius A, six, the correction term constant capital A. Concerning the first two of these factors, little need be said, unless a question be raised as to whether the density of such minute oil drops might not be a function of the radius. Such a question is conclusively answered in the negative, both by theory and by the experiments reported in this paper. Liquid rather than solid spheres were originally chosen because of the far greater certainty with which their density and sphericity could be known. Nevertheless, I originally used liquids of widely different viscosities, like oil, glycerin, mercury, and obtained the same results with them all within the limits of error, thus showing experimentally that, so far as this work was concerned, the drops all acted like rigid spheres. More complete proof of this conclusion is furnished both by the following observations and by other careful work on solid spheres soon to be reported in detail by Mr. J. Y. Lee. The material used for the drops in the following experiments was the highest grade of clock oil, the density of which, at 23 degrees centigrade, the temperature at which the experiments were carried out, was found in two determinations made four months apart to be 0.9199 with an error of not more than one part in 10,000. The electric fields were produced by a 5,300 volt storage battery, the PD of which dropped on an average 5 or 10 volts during an observation of an hour's duration. The potential readings were taken just before and just after a set of observations on a given drop by dividing the bank into six parts and reading the PD of each part with a 900 volt Kelvin and white electrostatic voltmeter, which showed remarkable constancy and could be read easily in this part of the scale, with an accuracy of about one part in 2,000. As a matter of fact, 5,000 volt readings made with the aid of two different calibration curves of the KNW instrument made two years apart never differed by more than one or two parts in 5,000. The value of capital F involves in addition to PD the distance between the plates, which was as before 16 millimeters and correct to about 0.01 millimeters, 1C page 351. Nothing more need be said concerning the first two of the above mentioned factors. The last four, however, need a special consideration. Section two, the coefficient of viscosity of air. This factor certainly introduces as large an element of uncertainty as inheres anywhere in the oil drop method. Since it appears in equation one in the three halves power, an uncertainty of 0.5% in eta means an uncertainty of 0.75% in E. It was therefore of the utmost importance that eta be determined with all possible accuracy. Accordingly, two new determinations were begun three years ago in the Ryerson Laboratory, one by Mr. Lachlan Gilchrist and one by Mr. I.M. Rapp. Mr. Gilchrist, whose work has already been published, used a constant deflection method with concentric cylinders, which it was estimated 1C page 386 ought to reduce the uncertainty in eta to one or two tenths of a percent. The results have justified this estimate. Mr. Rapp used a form of the capillary tube method, which it was thought was better adapted to an absolute evaluation of eta than have been the capillary tube arrangements which have been commonly used here to fore. Since Mr. Gilchrist completed his work at the University of Toronto, Canada, and Mr. Rapp made his computations and final reductions at Ersonist College, Pennsylvania, neither observer had any knowledge of the results obtained by the other. The two results agree within one part in 600. Mr. Rapp estimates his maximum uncertainty at 0.1%, Mr. Gilchrist at 0.2%. Mr. Rapp's work was done at 26 degrees centigrade and gave eta sub-26 equals 0.00018375. When this is reduced to 23 degrees centigrade, the temperature used in the following work by means of a formula 5, a formula which can certainly introduce no appreciable error for the range of temperature here used, viz equation 5, eta sub-t equals 0.00018240 minus 0.000000493 open parenthesis 23 minus t closed parenthesis. The results, eta sub-23 equals 0.00018227. Mr. Gilchrist's work was done at 20.2 degrees centigrade and gave eta sub-20.2 equals 0.0001812. When this is reduced to 23 degrees centigrade, it yields eta sub-23 equals 0.00018257. When this new work by totally dissimilar methods is compared with the best existing determinations by still other methods, the agreement is exceedingly striking. Thus in 1905, Hogg made at Harvard very careful observations on the damping of oscillating cylinders and obtained in three experiments at atmospheric pressure, eta sub-23 equals 0.0001825. Eta sub-15.6 equals 0.0001790 and eta sub-18.6 equals 0.0001795. These last two, reduced to 23 degrees centigrade as above, are 0.0001826 and 0.0001817 respectively. And the mean value of the three determinations is eta sub-23 equals 0.00018227. Tomlinson's classical determination, by far the most reliable of the 19th century, yielded when the damping was due primarily to push, eta sub-12.65 degrees centigrade equals 0.00017746. When it was wholly due to drag, eta sub-11.79 degrees centigrade equals 0.00017711. These values, reduced to 15 degrees centigrade as above, are respectively 0.00017862 and 0.00017867. Hence, we may take Tomlinson's direct determination as eta sub-15 equals 0.00017864. This reduced to 23 degrees centigrade by Tomlinson's own temperature formula, Ulmans, yields eta sub-23 equals 0.00018242. By the above formula, it yields eta sub-23 equals 0.00018256. Grindley and Gibson, using the tube method on so large a scale, tube 1.8 inch in diameter and 108 feet long, as to largely eliminate the most fruitful sources of error in this method, namely the smallness and uniformity of the bore, obtained at room temperature the following results. Eta sub-25.28 degrees centigrade equals 0.00018347. Eta sub-23.55 degrees centigrade equals 0.00018241. Eta sub-12.18 degrees centigrade equals 0.00018257. And Eta sub-15.4 degrees centigrade equals 0.0001782. These numbers reduced to 23 degrees centigrade as above are respectively 18.245, 18.241, 18.201, and 18.195. The mean is 18.220. Grindley and Gibson's own formula, eta equals 0.0001702. Open brace 1 plus 0.00329t minus 0.00070t2, close brace, yields eta sub-23 equals 0.00018245. We may take then Grindley and Gibson's direct determination as the mean of these two values. Viz eta sub-23 equals 0.00018232. Collecting then the five most careful determinations of the viscosity of error which so far as I am able to discover have ever been made, we obtain the following table. It will be seen then that every one of the five different methods which have been used for the absolute determination of eta leads to a value which differs by less than one part in 1,000 from the above mean value eta sub-23 equals 0.00018240. It is surely legitimate then to conclude that the absolute value of eta for error is now known with an uncertainty of somewhat less than one part in 1,000. A second question which might be raised in connection with eta is as to whether the medium offers precisely the same resistance to the motion through it of a heavily charged drop as to that of an uncharged drop. This question has been carefully studied and definitely answered in the affirmative by the following work. Compare sections 6 and 10. Section 3, the speeds V sub-1 and V sub-2. The accuracy previously attained in the measurement of the times of ascent and descent between fixed crosshairs was altogether satisfactory. But the method which had to be employed for finding the magnifying power of the optical system, i.e. for finding the actual distance of fall of the drop in centimeters, left something to be desired. This optical system was before a short focus telescope of such depth of focus that it was quite impossible to obtain an accurate measure of the distance between the crosshairs by simply bringing a standard scale into sharp focus immediately after focusing upon a drop. Accordingly, as stated in the original article, the standard scale was set up at the exact distance from the telescope of the pinhole through which the drop entered the field. This distance could be measured with great accuracy, but the procedure assumed that the drop remained exactly at this distance throughout the whole of any observation, sometimes of several hours duration. But if there were the slightest lack of parallelism between gravity and the lines of the electric field, the drop would be obliged to drift slowly and always in the same direction away from this position. And a drift of 5mm was enough to introduce an error of 1%. Such a drift could in no way be noticed by the observer if it took place in the line of sight. For the speeds of the drops were changing very slowly anyway because of evaporation, fall in the potential of the battery, etc., and a change in time due to such a drift would be completely mashed by other causes of change. This source of uncertainty was well recognized at the time of the earlier observations, and steps were taken at the beginning of the present work to eliminate it. It was in fact responsible for an error of nearly 2%. A new optical system was built, consisting of an achromatic objective of 28mm aperture and 12.5cm focal length and an eyepiece of 12mm focal length. The whole system was mounted in a support, which could be moved bodily back and forth by means of a horizontal screw of 1.5mm pitch. In an observation, the objective was 25cm distant from the drop, which was kept continually in sharp focus by advancing or withdrawing the whole telescope system. The depth of focus was so small that a motion of 1.5mm blurred badly the image of the drop. The eyepiece was provided with a scale having 80 horizontal divisions, and the distance between the extreme divisions of this scale, the distance of fall in the following experiments, could be regularly duplicated with an accuracy of at least one part in 1000 by bringing a standard scale, Société Genovois, into sharp focus. The optical path when the scale was viewed was made exactly the same as when the drop was viewed. The distance of fall, then, one of the most uncertain factors of the preceding determination, was now known with at least this degree of precision. The accuracy of the time determinations can be judged from the data in tables 4 through 19. On account of the great convenience of a direct reading instrument, these time measurements were all made, not with a chronograph as heretofore, but with a hip chronoscope, which read to 0.002 seconds. This instrument was calibrated by comparison with the standard Vyerson Laboratory Clock under precisely the same conditions as those under which it was used in the observations themselves, and found to have an error between 0 and 0.2%, depending upon the time interval measured. For the sake of enabling others to check all the computations herein contained if desired, as well as for the sake of showing what sort of consistency was obtained in the measurement of time intervals, there are given in Table 2 the calibration readings for the 30 second interval, and in Table 3 the result of similar readings for all the intervals used. The change in the percent correction with the time interval employed is due to the difference in the reaction times of the magnet and spring content at make, beginning, and at break, end. All errors of this sort are obviously completely eliminated by making the calibration observations under precisely the same conditions as the observations on the drop. In Tables 4 to 19, the recorded times are the uncorrected chronograph readings. The corrections are obtained by interpolation in the last column of Table 3. Under the head of possible uncertainties in the velocity determinations are to be mentioned also the effects of a distortion of the drop by the electric field. Such a distortion would increase the surface of the drop, and hence the speed imparted to it per dime of electric force would not be the same as the speed imparted per dime of gravitational force when the field was off and the drop had the spherical form. The following observations were made in such a way as to bring to light such an effect if it were of sufficient magnitude to exert any influence whatever upon the accuracy of the determination of E by this method. Compare sections 6 and 10. Similarly, objection has been made to the oil drop method on the ground that, on account of internal convection, fluid drops would not move through air with the same speed as solid drops of light diameter and mass. Such objection is theoretically unjustifiable in the case of oil drops of the sizes here considered. Nevertheless, the experimental demonstration of its invalidity is perhaps worthwhile and is therefore furnished below. Section 4, the radius A. The radius of the drop enters only into the correction term, C equation 4, and so long as this is small, need not be determined with a high degree of precision. It is most easily obtained by the following procedure, which differs slightly from that originally employed, 1C page 379. It will be seen that the equation 1C page 353, equation 6, V sub 1 over V sub 2 equals fraction numerator mg denominator capital F E minus mg and fraction. Contains no assumption whatever save that a given body moves through a given medium with a speed which is proportional to the force acting upon it. Substitution in this equation of M equals 4 thirds pi A cubed open parenthesis sigma minus rho closed parenthesis. And the solution of the resulting equation for A gives equation 7. A equals cube root radical fraction numerator 3 capital F E denominator 4 pi g open parenthesis sigma minus rho closed parenthesis and fraction. Fraction numerator V sub 1 denominator V sub 1 plus V sub 2 and fraction and radical. The substitution in this equation of an approximately correct value of E yields A with an error but one third as great as that contained in the assumed value of E. The radius of the drop can then be determined from 7 with a very high degree of precision as E becomes more and more accurately known. In the following work the value of E substituted in 7 to obtain A was 4.78 times 10 to the negative tenth power. But the final value of E obtained would not have been appreciably different if the value substituted in 7 to obtain A had been 5% or 6% in error. The determination of A therefore introduces no perceptible error into the evaluation of E. End of sections 1 through 4. Sections 5 through 8 of On the elementary electrical charge and the Avogadro constant. Physical review volume 2 number 2 by Robert Andrews Millican. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Section 5. The correction term constant A. This constant was before graphically determined 1C page 379 by plotting the values of E sub 1 to the two thirds power as ordnance and those of L over A as abscissie. And observing that if we let X equal L over A, Y equal E sub 1 to the two thirds power and Y zero equal E to the two thirds power. Equation 3 may be written in the form. Equation 8. Y sub zero times open parenthesis 1 plus capital AX closed parenthesis equals Y. Or equation 9. Capital A equals DY DX over Y sub zero equals slope over Y intercept. Now even if the slope were correctly determined by the former observations, all of the above mentioned sources of error would enter into the value of the intercept and hence would modify the value of capital A. As a matter of fact however the accuracy with which the slope itself was determined could be much improved. For with the preceding arrangement it was necessary to make all the observations at atmospheric pressure. And the only way of varying L over A was by varying A, i.e. by using drops of different radii. But when A was very small, the drops moved exceedingly slowly under gravity and the minutest of residual convection currents produced relatively large errors in the observed speeds, i.e. in E sub 1. If for example the time of fall over a distance of 2mm is 20 minutes, it obviously requires an extraordinary degree of stagnancy to prevent a drift in that time of say 0.2mm due to convection. But this would introduce an error of 10% into E sub 1. Furthermore with these slow drops Brownian movements introduce errors which can only be eliminated by taking a very large number of readings, and this is not in general feasible with such drops. It is quite impossible then by working at a single pressure to obtain from the graph mentioned above a line long enough, 1C page 379, to make the determination of its slope a matter of great precision. Accordingly in the new observations the variation of L over A was affected chiefly through the variation of L, i.e. of the pressure P, rather than of A. This made possible not only the accurate evaluation of E, but also the solution of the interesting equation as to the law of fall of a given drop through air at reduced pressures. Section 6. Method of testing the assumptions involved in the oil drop method. In order to make clear the method of treatment of the following observations, a brief consideration of the assumptions underlying the oil drop method must here be made. These assumptions may be stated thus. 1. The drag which the medium exerts upon a given drop is unaffected by its charge. 2. Under the conditions of observation, the oil drops move through the medium essentially as solid spheres. This assumption may be split into two parts and stated thus. Neither to A, distortions due to the electric field, nor to B, internal convection within the drop, modify appreciably the law of motion of an oil drop. 3. The density of oil droplets is independent of their radius, down to A equals 0.00005 cm. Of these assumptions, 2A is the one which needs the most careful experimental test. It will be seen that it is contained in the fundamental equation of the method C7, which may be written in the form Equation 10. E sub n equals fraction numerator mg denominator capital F v sub 1 n fraction times open parenthesis v sub 1 plus v sub 2 closed parenthesis. Or, still more conveniently, in the form Equation 11. E sub n equals fraction mg t sub g over capital F n fraction open parenthesis 1 over t sub g plus 1 over t sub capital F closed parenthesis. In which t sub g and t sub capital F are the respective time intervals required by the drop to fall under gravity and to rise under the field capital F, the distance between the crosshairs. In order to see how the assumption under consideration can be tested, let us write the corresponding equation after the same drop has caught n prime additional units. Namely, Equation 12. E sub n plus n prime equals fraction mg t sub g over capital F n fraction times open parenthesis 1 over t sub g plus 1 over t sub capital F prime closed parenthesis. The subtraction of 11 from 12 gives Equation 13. E sub n prime equals fraction mg t sub g over capital F n fraction times open parenthesis 1 over t sub capital F prime minus 1 over t sub capital F n parenthesis. Now, Equations 11 and 12 show, since mg t sub g over capital F remains constant, that as the drop changes charge, the successive values of its charge are proportional to the successive values assumed by the quantity 1 over t sub g plus 1 over t sub capital F. And the elementary charge itself is obviously the same constant factor, mg t sub g over capital F, multiplied by the greatest common divisor of all these successive values. It is to be observed, too, that since 1 over t sub g is in these experiments generally large compared to 1 over t sub F, the value of this greatest common divisor, which will be noted by open parenthesis 1 over t sub g plus 1 over t sub F and parenthesis sub 0, is determined primarily by the time of fall under gravity, and is but little affected by the time in the field. On the other hand, Equation 13 shows that the greatest common divisor of the various values of 1 over t sub F prime minus 1 over t sub F, which will be designated by open parenthesis 1 over t sub F prime minus 1 over t sub F, closed parenthesis sub 0, when multiplied by the same constant factor, mg t sub g over capital F, is also the elementary electrical charge. In other words, open parenthesis 1 over t sub g plus 1 over t sub F, closed parenthesis sub 0, and open parenthesis 1 over t sub F prime minus 1 over t sub capital F, closed parenthesis sub 0, are one and the same quantity. But while the first represents essentially a speed measurement when the field is off, the second represents a speed measurement in a powerful electric field. If then the assumption under consideration is correct, we have two independent ways of obtaining the quantity, which when multiplied by the constant factor, mg t sub g over capital F, is the elementary electrical charge. But if on the other hand, the distortion of the drop by the field modifies the law of motion of the oil drop through the medium, then open parenthesis 1 over t sub g plus 1 over t sub capital F, closed parenthesis sub 0, and open parenthesis 1 over t sub capital F prime minus 1 over t sub capital F, closed parenthesis sub 0, will not be the same. Now a very careful experimental study of the relations of open parenthesis 1 over t sub g plus 1 over t sub capital F, closed parenthesis sub 0, and open parenthesis 1 over t sub capital F prime minus 1 over t sub capital F, closed parenthesis sub 0, shows so perfect agreement that no effect of distortion in changing measurably the value of E can be admitted. See tables 4 to 14. Turning next to assumption 1, this can be tested in three ways, all of which have been tried with negative results. First, a drop containing from 1 up to 6 or 7 elementary charges can be completely discharged and its time of fall under gravity when uncharged, compared with its time when charged. Second, the multiple relationships shown in the successive charges carried by a given drop may be very carefully examined. They cannot hold exactly if when the drop is heavily charged it suffers a larger drag from the medium than when it is lightly charged. Third, when drops having widely different charges and different masses are brought to the same value of L over A by varying the pressure, the value of E sub 1, which is proportional to open parenthesis V sub 1 plus V sub 2, closed parenthesis sub 0, should come out smaller for the heavily than for the lightly charged drops. The following observations show that this is not the case. The last criterion is also a test for 2B, for if internal convention modifies the speed of fall of a drop as Perron wishes to assume that it may, it must play a smaller and smaller role as the drop diminishes in size, hence varying L over A by diminishing A cannot be equivalent to varying L over A by increasing L. In other words, the value of E sub 1 obtained from work on a large drop at a low pressure should be different from that obtained from work on a small drop at so high a pressure that L over A has the same value as for the large drop. Finally, if the density of a small drop is greater than that of a large one, C assumption 3, then for a given value of L over A, the small drop will show a larger value of E sub 1 than the large one in as much as the computation of E sub 1 is based on a constant value of sigma. The fact then that for a given value of L over A, the value of E sub 1 actually comes out independent of the radius or charge of the drops shows conclusively either that no one of these possible sources of error exists or else that they neutralize one another so that for the purposes of this experiment they do not exist. That they do not exist at all is shown by the independent theoretical and experimental tests mentioned above. This removes, I think, every criticism which has been suggested of the oil drop method of determining E and N. Section 7. Summary of improvements in method. In order to obtain the consistency shown in the following observations, it was found necessary to take much more elaborate precautions to suppress convection currents in the air of the observing chamber than had at first been thought needful. To recapitulate then the improvements which have been introduced into the oil drop method consist in 1. A redetermination of eta. 2. An improved optical system. 3. An arrangement for observing speeds at all pressures. 4. The more perfect elimination of convection. 5. The experimental proof of the correctness of all the assumptions underlying the method. Viz. A. That a charge does not alter the drag of the medium on the charged body. B. That the oil drops act essentially like solid spheres. C. That the density of the oil drops is the same as the density of the oil in bulk. Section 8. The experimental arrangements. The experimental arrangements are shown in figure 1. The brass vessel capital D was built for work at all pressures up to 15 atmospheres. But since the present observations have to do only with pressures from 76 cm down, these were measured with a very carefully made mercury manometer M, which at atmospheric pressure gave precisely the same reading as a standard barometer. Complete stagnancy of the air between the condenser plates capital M and capital N was attained first by absorbing all of the heat rays from the arc capital A by means of a water cell W, 80 cm long, and a cupric chloride cell D, and second by immersing the whole vessel capital D in the constant temperature bath capital G of gas engine oil, 40 l, which permitted in general fluctuations of not more than .02 degrees centigrade during an observation. This constant temperature bath was found essential if such consistency of measurement as is shown below was to be obtained. A long search for causes of slight irregularity revealed nothing so important as this, and after the bath was installed all of the irregularities vanished. The atomizer capital A was blown by means of a puff of carefully dried and dust free air introduced through the cock E. The air about the drop P was ionized when desired by means of Rentsgen rays from capital X, which readily passed through the glass window G. To the three windows G, two only are shown. In the brass vessel capital D correspond of course three windows in the ebonite strip C, which encircles the condenser plates capital M and capital N. Through the third of these windows, set at an angle of about 18 degrees from the line capital X P A, and in the same horizontal plane the oil drop is observed. End of sections five through eight. Sections nine through thirteen of on the elementary electrical charge and the Avogadro constant. Physical review volume two number two by Robert Andrews Millican. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Section nine the observations. The record of a typical set of readings on a given drop is shown in table four. The first column headed T sub G gives the successive readings on the time of descent under gravity. The fourth column headed T sub capital F sub C gives the successive times of ascent under the influence of the field capital F as measured on the hip chronoscope. These two columns contain all the data which is used in the computations. But in order to have a test of the stagnancy of the air a number of readings were also made with a stopwatch on the times of ascent through the first half and through the whole distance of ascent. These readings are found in the second and third columns the times for the first half under the head one half T sub capital F sub G. The times for the whole distance under the head T sub capital F sub G. It will be seen from these readings that there is no indication whatever of convection since the readings for the one half distance have uniformly one half the value of the readings for the whole distance within the limits of error of a stopwatch measurement. This sort of a test was made on the majority of the drops but since no further use is made of these stopwatch readings they will not be given in succeeding tables. The fifth column headed one over T sub capital F contains the reciprocals of the values in the fourth column after the correction found from tables two and three has been applied. The sixth column contains the successive differences in the values of one over T sub F resulting from the capture of ions. The seventh column headed N prime contains the number of elementary units caught at each change a number determined simply by observing what number the quantity just before it in column six must be divided to obtain the constancy shown in the eighth column which contains the successive determinations of open parenthesis one over T sub capital F prime minus one over T sub capital F close parenthesis sub zero C section six. Similarly, the ninth column headed N gives the total number of units of charge on the drop a number determined precisely as in the case of the numbers in the seventh column by observing by what numbers the successive values of one over T sub G plus one over T sub capital F must be divided to obtain the constancy shown in the tenth column which contains the successive values of open parenthesis one over T sub G plus one over T sub capital F close parenthesis sub zero. Since N prime is always a small number and in some of the changes almost always has the value one or two, its determination for any change is obviously never a matter of the slightest uncertainty. On the other hand, N is often a large number but with the aid of the known values of N prime it can always be found with absolute certainty so long as it does not succeed say 100 or 150. It will be seen from the means at the bottom of the eighth and the tenth columns that in the case of this drop the two ways discussed in section six of obtaining the number which when multiplied by M G T sub G over capital F is the elementary electrical charge yield absolutely identical results. In order to show the sort of consistency attained in this work the complete records are given in tables five, six, seven, eight and nine of five drops of practically the same size taken at very nearly the same pressures. These are the only drops of this size which were observed with the present arrangement so that they represent the regular run of observations rather than the selected group. The symbols in the last column capital V sub I, capital V sub F, T, P, V sub 1 and A stand for initial volts, final volts, temperature, pressure, velocity under gravity and radius respectively. It will be seen from the second columns, C table five, that in general in spite of the precautions taken against evaporation one C page 388 the drops do evaporate very slowly since with a given charge the speed in the field very slowly increases. It is for this reason that the numbers in the next to last column tend to increase very slightly. This slow change introduces no error into E sub 1 provided corresponding values of T sub G, T sub capital F and capital V volts are combined to obtain E sub 1. The mean values taken throughout the same time interval constitute such corresponding values. On account of this slow change however the readings from which the differences 1 over T sub capital F prime minus 1 over T sub capital F are taken should be separated by as small a time interval as possible. It is for this reason that in table five for example to obtain the difference 1 over T sub capital F prime minus 1 over T sub capital F corresponding to the change from T sub capital F equals 45 seconds to T sub capital F equals 694 seconds. The last two of the 45 second readings are averaged instead of the whole four. It is worthy to note too in this connection that if a change in the time in the field takes place first from 45 seconds to 694 seconds and then immediately back again to the 45 second value. The mean of the two values of 1 over T sub capital F prime minus 1 over T sub capital F thus obtained is independent of any error arising from the evaporation of the drop. For this reason the mean value of the quantity fraction 1 over N prime N fraction times open parenthesis 1 over T sub capital F prime minus 1 over T sub capital F is in general much more trustworthy than might be inferred from the variation in the individual numbers from which this mean is obtained. Nevertheless, in practically all of the following work in view of the large number of observations in the T sub G column, the mean at the bottom of the column 1 over N times open parenthesis 1 over T sub G plus 1 over T sub capital F closed parenthesis is considered more trustworthy than the mean at the bottom of the column 1 over N prime times open parenthesis 1 over N prime. The mean at the bottom of the column 1 over T sub capital F prime minus 1 over T sub capital F closed parenthesis and it has been in fact exclusively used in the computation of E sub 1 only in the case of two or three of the fastest drops tables 16 and 17 are the closest parenthesis 1 over T sub G plus 1 over T sub capital F closed parenthesis sub 0 greater perhaps than those in open parenthesis 1 over T sub capital F prime minus 1 over T sub capital F closed parenthesis sub 0. And in these cases the two were so near together that there was no object in using the latter instead of the former. It should be stated that all time intervals longer than 150 seconds were measured with the stopwatch rather than with the chronoscope and are not subjected to the corrections in tables two and three. In general two only differences in T sub capital F prime minus T sub capital F amounting to as much as 20 seconds are used in the determination of 1 over T sub capital F prime minus 1 over T sub capital F. Since obviously the observational error is large when T sub capital F prime minus T sub capital F is small. A study of tables 4 to 9 shows first a striking agreement between the values of open parenthesis 1 over T sub capital F prime minus 1 over T sub capital F closed parenthesis sub 0 and open parenthesis 1 over T sub G plus 1 over T sub capital F closed parenthesis sub 0. The largest divergence being found in table 5 where it amounts to 0.3%. And second a satisfactory agreement between the values of E sub 1 obtained from the different drops of approximately the same size the largest divergence of mean value being here to 0.3%. A few typical records of observations made at different pressures C P last column and on drops of different sizes C A last column are presented in tables 10 to 19. Table 20 contains a complete summary of the results obtained on all of the 58 different drops upon which complete series of observations like the above were made during a period of 60 consecutive days. It will be seen from this table that these observations represent a 30 fold variation in L over A from 0.016 drop number 1 to 0.444 drop number 58. A 17 fold variation in P from 4.46 centimeters drop number 56 to 76.27 centimeters drop number 10. A 12 fold variation in A from 4.69 times 10 to the negative fifth power centimeters drop number 28 to 58.56 times 10 to the negative fifth centimeters drop number 1. And a variation in the number of free electrons carried by the drop from 1 on drop number 28 to 136 on drop number 56. The time of fall of drop number 28 was also tested when it was completely discharged as have been the times of many other drops which carried most of the time but one electron. Much larger variations both in A and P and therefore in L over A might have been used and have in fact been used for finding the law of fall of a drop through rarefied air. But for the end here sought, namely the most accurate possible determination of E, it was found desirable to keep the T sub G interval for the most part between the limits 10 seconds and 40 seconds in order to avoid chronograph errors on the one hand and Brownian movement irregularities on the other. That neither of these sources of error is appreciable in these observations may be seen from a study of tables 4 through 19 which are thoroughly representative of the work on all the drops. Section 10. Results and discussion. It will be seen at once from equation 4 that the value of E is simply the value of E sub 1 for which L over A equals 0. So that if successive values of E sub 1 to the two-thirds power are plotted as obssessy and of L over A as ordinance, the intercept of the resulting curve on the E sub 1 to the two-thirds power axis is E to the two-thirds power. Furthermore, if capital A is a constant, then the curve in question is a straight line, and A is the slope of this line divided by the Y intercept, C equation 9. In view of the uncertainty in L, due to the fact that K in the equation A equals KNM C bar L has never been exactly evaluated, it was not preferable to write the correction term to Stokes' law, C 2 and 3. In the form, open parenthesis 1 plus B over PA, closed parenthesis to the negative 1 power, instead of in the form, open parenthesis 1 plus capital A L over A, closed parenthesis to the negative 1 power, and then to plot E sub 1 to the two-thirds against 1 over PA. Nevertheless, in view of the greater ease of visualization of L over A, all the values of this quantity corresponding to successive values of 1 over PA are given in table 20, K being taken merely for the purposes of this computation as 0.3502, Boltzmann. Figure 2 shows the graph obtained by plotting the values of E sub 1 to the two-thirds against 1 over PA for the first 51 drops of table 20, and Figure 3 shows the extension of this graph to twice as large values of 1 over PA and E sub 1 to the two-thirds power. It will be seen that there is not the slightest indication of a departure from a linear relation between E sub 1 to the two-thirds power and 1 over PA up to the value 1 over PA equals 620.2, which corresponds to a value of L over A of 0.4439, C drop number 58 table 20. Furthermore, the scale used in the plotting is such that a point which is one division above or below the line in Figure 2 represents in the mean an error of 2 and 700. It will be seen from figures 2 and 3 that there is but one drop in the 58 whose departure from the line amounts to as much as 0.5%. It is to be remarked, too, that this is not a selected group of drops, but represents all of the drops experimented upon during 60 consecutive days, during which time the apparatus was taken down several times and set up anew. It is certain, then, that an equation of the form 2 holds very accurately up to L over A equals 0.4. The last drop in Figure 3 seems to indicate the beginning of a departure from this linear relationship. Since such departure has no bearing upon the evaluation of E, discussion of it will be postponed to another paper. Attention may also be called to the completeness of the answers furnished by figures 2 and 3 to the questions raised in Section 6. Thus, drops number 27 and 28 have practically identical values of 1 over PA, but while number 28 carries during part of the time but one unit of charge, C table 20, drop number 27 carries 29 times as much, and it has about 7 times as large a diameter. Now if the small drop were denser than the large one, see Assumption 3, Section 6, or if the drag of the median upon the heavily charged drop were greater than its drag upon the one lightly charged, see Assumption 1, Section 6, then for both these reasons, drop 27 would move more slowly relative to drop 28 than would otherwise be the case. And hence E sub 1 to the two-thirds power for 27 would fall below E sub 1 to the two-thirds power for drop 28. Instead of this, the two E sub 1 to the 23rd powers fall so nearly together that it is impossible to represent them on the present scale by two separate dots. Drops 52 and 56 furnish an even more striking confirmation of the same conclusion, for both drops have about the same value for L over A, and both are exactly on the line, though number 56 carries at one time 68 times as heavy a charge as number 52 and has three times as large a radius. In general, the fact that figures 2 and 3 show no tendency whatever on the part of either the very small or the very large drops to fall above or below the line is experimental proof of the joint correctness of Assumptions 1, 3, and 2B of Section 6. The correctness of 2A was shown by the agreement throughout Tables 4 to 19 between 1 over N prime times open parenthesis 1 over T sub capital F prime minus 1 over T sub capital F closed parenthesis, and 1 over N times open parenthesis 1 over T sub G plus 1 over T sub capital F closed parenthesis. The values of E to the two-thirds power and B obtained graphically from the Y intercept and the slope in Figure 2 are E to the two-thirds power equals 61.13 times 10 to the negative eighth power and B equals .0006254. P being measured for the purposes of Figure 2 and of this computation in millimeters of mercury at 23 degrees centigrade and A being measured in centimeters. The value of capital A equations 2 and 3 corresponding to this value of capital B is .874 instead of .817 as originally found. Cunningham's theory gives in terms of the constants here used capital A equals 788. Instead, however, of taking the result of this graphical evaluation of E to the two-thirds power, it is more accurate to reduce each of the observations on E sub 1 to the two-thirds power to E to the two-thirds power by means of the above value of capital B and the equation, equation 14. E to the two-thirds power times open parenthesis 1 plus B over PA closed parenthesis equals E sub 1 to the two-thirds power. The results of this reduction are contained in the last column of Table 20. These results illustrate very clearly the sort of consistency obtained in these observations. The largest departure from the mean value found anywhere in the table amounts to 0.5%, and the probable error of the final mean value computed in the usual way is 16 and 61,000. Instead, however, of using this final mean value as the most reliable evaluation of E to the two-thirds power, it was not preferable to make a considerable number of observations at atmospheric pressure, on drops small enough to make T sub G determinable with great accuracy, and yet large enough so that the whole correction term to Stokes law amounted to but a few percent. Since in this case, even though there might be a considerable error in the correction term constant B, such error would influence the final value of E by an inappreciable amount. The first 23 drops of Table 20 represent such observations. It will be seen that they show slightly greater consistency than do the remaining drops in the table, and that the correction term reductions for these drops all lie between 1.3%, drop number 1, and 5.6%, drop number 23. So that even though B were in error by as much as 3%, its error is actually not more than 0.5%, E to the two-thirds power would be influenced by that fact to the extent of but 0.1%. The mean value of E to the two-thirds power obtained from the first 23 drops is 61.12 times 10 to the negative 8th power, a number which differs by one part in 3400 from the mean obtained from all the drops. When correction is made for the fact that the numbers in Table 20 were obtained on the basis of the assumption A to the sub-23 equals 0.0001825, instead of A to the sub-23 equals 0.0001824, C section 2, The final mean value of E to the two-thirds power obtained from the first 23 drops is 61.085 times 10 to the negative 8th power. This corresponds to E equals 4.774 times 10 to the negative 10th power electrostatic units. Since the value of the Faraday constant has now been fixed virtually by international agreement at 9,650 absolute electromagnetic units, and since this is the number N of molecules in a gram molecule times the elementary electrical charge, we have N times 4.774 times 10 to the negative 10th power equals 9,650 times 2.9990 times 10 to the 10th power. Therefore, N equals 6.062 times 10 to the 23rd power. Although the probable error in this number computed by the method of least squares from Table 20 is but one part in 3000, it would be erroneous to infer that E and N are now known with that degree of precision. For there are four constant factors entering into all of the results in Table 20 and introducing uncertainties as follows. The coefficient of viscosity, eta, which appears in the three-halves power introduces into E and N, a maximum possible uncertainty of 0.1%. The distance between the condenser plates, 16.00 mm, is correct to 0.01 mm, and therefore since it appears in the first power in E introduces a maximum possible error of something less than 0.1%. The voltmeter readings have a maximum possible error of rather less than 0.1%, and carry this in the first power into E and N. The crosshair distance, which is uniformly duplicatable to one part in a thousand, appears in the three-halves power and introduces an uncertainty of no more than 0.1%. The other factors introduce errors which are negligible in comparison. The uncertainty in E and N is then that due to four continuous factors, each of which introduces a maximum possible uncertainty of 0.1%. Following the usual procedure, we may estimate the uncertainty in E and N as the square root of the sum of the squares of these four uncertainties, that is, as two parts in one thousand. We have then finally E equals 4.774 plus or minus 0.009 times 10 to the negative tenth power, and N equals 6.062 plus or minus 0.012 times 10 to the 23rd power. The difference between these numbers and those originally found by the oil drop method, vis E equals 4.891, and N equals 5.992, is due to the fact that this much more elaborate and prolonged study has had the effect of changing every one of the three factors, eta, capital A, and D, crosshair distance, in such a way as to lower E and to raise capital N. The chief's change, however, has been due to the elimination of the faults of the original optical system. Section 11, comparison with other measurements. So far as I am aware, there is at present no determination of E or capital N by any other method which does not involve an uncertainty at least 15 times as great as that represented in the above measurements. Thus the radioactive method yields in the hands of Regener, account of the alpha particles, which gives E with an uncertainty which E estimates at 3%. This is as high a precision, I think, as has yet been claimed for any alpha particle count, though Geiger and Rutherford's photographic registration method will doubtless be able to improve it. The Brownian movement method yields results which fluctuate between Perrin's value, E equals 4.24 times 10 to the negative tenth power, and Fletcher's value, 5.01 times 10 to the negative tenth power, with Svedberg's measurements yielding the intermediate number, 4.7 times 10 to the negative tenth power. The radiation method of Planck yields capital N as a product of C sub 2 to the third power and sigma. The latest Regsenstalt value of C sub 2 is 1.436 to the eighth power, while Koblenz as the result of extraordinarily careful and prolonged measurements obtains 1.4456. The difference in these two values of C sub 2 to the third power is 2%. Westfall estimates his error in the measurement of sigma at 0.5%, though reliable observers differ in it by 5% or 6%. We may take then 3% as the limit of accuracy thus far attained in measurements of E or capital N by other methods. The mean results by each one of the three other methods fall well within this limit of the value found above by the oil drop method. Section 12, computation of other fundamental constants. For the sake of comparison and reference, the following fundamental constants are recomputed on the basis of the above measurements. 1. The number N of molecules in 1cc of an ideal gas at 0°76 is given by N equals capital N over capital V, which equals 6.062 times 10 to the 23rd over 22,412, which equals 2.705 times 10 to the 19th power. 2. The mean kinetic energy of agitation, capital E sub 0 of a molecule at 0° centigrade, is given by P capital V equals one-third capital NMU squared, which equals two-thirds capital N capital E0, which equals capital R capital T. Therefore, capital E0 equals three-halves P sub 0 V sub 0 over capital N, which equals fraction numerator 3 times 1,013,700 times 22,412, denominator 2 times 6.062 times 10 to the 23rd, and fraction, which equals 5.621 times 10 to the negative 14th power ergs. 3. The constant epsilon of molecular energy defined by capital E sub 0 equals epsilon capital T is given by epsilon equals capital E sub 0 over capital T, which equals 5.621 times 10 to the negative 14th power over 273.11, which equals 2.058 times 10 to the negative 16th power ergs over degrees. 4. The Boltzmann entropy constant K, defined by capital S equals K log capital W, is given by K equals capital R over capital N, which equals P sub 0 capital V sub 0 over capital T capital N, which equals two-thirds epsilon, which equals 1.372 times 10 to the negative 16th power ergs over degrees. All of these constants are known with precisely the accuracy attained in the measurement of E. 5. The Planck-Wierkungsquantum H can probably be obtained considerably more accurately as follows than in any other way. From equation 292, page 166 of the Verme-Strahlung, we obtain H equals K to the four-thirds power over C times open parenthesis 48 pi alpha over A closed parenthesis to the one-third power, which equals fraction numerator open parenthesis 1.372 times 10 to the negative 16th power closed parenthesis to the four-thirds power, denominator 2.999 times 10 to the 10th power and fraction times open parenthesis fraction numerator 48 pi times 1.0823 denominator 7.39 times 10 to the negative 15th power and fraction closed parenthesis to the one-third power, which equals 6.620 times 10 to the negative 27th power, which gives H with the same accuracy attainable in the measurement of K to the four-thirds power over A in which A is the Stefan-Boltzmann constant. If Westfall's estimate of his error in the measurement of this constant is correct, this 0.5%, it would introduce an uncertainty of but 0.2% into H. This is about that introduced by the above determination of K to the four-thirds. Hence, the above value of H should not be an error by more than 0.4%. 6. The constant C sub 2 of the Wien-Planck radiation law may also be computed with much precision from the above measurements. For also from equation 292 of the Wehrmastrahlung, we obtain C sub 2 equals open parenthesis 48 pi alpha K over A closed parenthesis to the one-third power, which equals open parenthesis fraction numerator 48 pi times 1.0823 times 1.372 times 10 to the negative 16th power, denominator 7.39 times 10 to the negative 15th power, end fraction closed parenthesis to the one-third power, which equals 1.4470 centimeter degrees. Since both K and A here appear in the one-third power, the error in C should be no more than 0.2%, provided Westfall's error is no more than 0.5%. The difference between this and Koblenz mean value, Viz 1.4456, is but 0.1%. The agreement is then entirely satisfactory. A further independent check is found in the fact that Day and Sossmann's location of the melting point of platinum at 1,755 degrees centigrade is equivalent to a value of C sub 2 equals 1.4475. On the other hand, the last Reichenstalt value of C sub 2, Viz 1.437, is too low to fit well with these and Westfall's measurements. It fits perfectly, however, with a combination of the above value of E and Shakespeare's value of sigma, Viz sigma equals 5.67 times 10 to the negative fifth power. Section 13, Summary. The results of this work may be summarized in the following table, in which the numbers in the error column represent the case of the first six numbers, estimated limits of uncertainty, rather than the so-called probable errors, which would be much smaller. The last two constants, however, involve Westfall's measurements and estimates and Planck's equations, as well as my own observations. Table 21, elementary electrical charge, E equals 4.774 plus or minus 0.009 times 10 to the negative tenth power. Number of molecules per gram molecule, capital N equals 6.062 plus or minus 0.012 times 10 to the 23rd power. Number of gas molecules per Cc at 0 degrees 76, N equals 2.705 plus or minus 0.005 times 10 to the 19th power. Kinetic energy of a molecule at 0 degrees centigrade, capital E sub zero equals 5.621 plus or minus 0.010 times 10 to the negative fourth power. Constant of molecular energy, epsilon equals 2.058 plus or minus 0.004 times 10 to the negative sixteenth power. Constant of the entropy equation, K equals 1.372 plus or minus 0.002 times 10 to the negative sixteenth power. Elementary Virkin's quantum, H equals 6.620 plus or minus 0.025 times 10 to the negative 27th power. Constant of the Vien displacement law, C sub 2 equals 1.4470 plus or minus 0030. I take pleasure in acknowledging the able assistance of Mr. J. Yen Bong Lee in making some of the above observations. Mr. Lee has also repeated with my apparatus the observations on oil at atmospheric pressure with results which are nearly as consistent as the above. Using my value of B, he obtains as a mean of measurements on 14 drops a value of E which differs from the above by less than one part and 6,000. Although it's probable error computed as in the case of table 20 is one part and 2,000. Ryerson physical laboratory, University of Chicago, June 2, 1913. End of sections 9 through 13. End of On the elementary electrical charge and the Avogadro constant. Physical review, volume 2, number 2 by Robert Andrews Millican.