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 The atom; 
 
 Cornell University Library 
 
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THE ATOM 
 
 NEW YORK 
 D. VAN NOSTRAND COMPANY 
 
 Eight Warren Street 
 1920 
 
COPYRIGHT, 1920, BY 
 D. VAN NOSTRAND COMPANY 
 
TO 
 
 MY MOTHER 
 
Preface 
 
 |HILE the present volume is in the main de- 
 voted to an exposition of a new theory of 
 the atom, the results which are incident to 
 ^^1 this theory have assumed an unexpected im- 
 portance. New theoretical values for Rydberg's con- 
 stant and Planck's constant have been obtained, from 
 which numerical values of all of the important constants' 
 connected with the electrons have been derived, namely 
 Planck's constant, the electronic charge, the masses of 
 the electron and hydrogen nucleus. These agree within 
 the limits of error with the direct experimental determi- 
 nation of these constants. 
 
 These new theoretical expressions have supplied the 
 missing equation by means of which the dimensions of 
 the two aetherial constants, specific inductive capacity 
 and magnetic permeability, become separately known, 
 a matter the importance of which has recently been 
 emphasized by Sir Oliver Lodge. Not only have the 
 dimensions of the two aetherial constants been found in 
 terms of length and of time, but those of ordinary mass 
 as well, so that the dimensions of all kinds of quantities, 
 — electrical, magnetic and mechanical, — are capable of 
 expression in terms of length and of time alone. A table 
 of dimensions has been prepared giving the dimensions of 
 the more common units in terms of length and of time, 
 referred to as the space-time system of units. New 
 units of length and of time are considered in place of the 
 centimeter and the second, as a result of which the im- 
 
vi Preface 
 
 portance of expressing the specific inductive capacity 
 or magnetic permeability in all electromagnetic equa- 
 tions is very apparent. The velocity of light with the 
 new units becomes numerically unity, and so does twice 
 the Rydberg constant. The specific inductive capacity 
 becomes numerically equal to 3 X lo'", the velocity of 
 light on the C.G.S. system of units, and to omit to ex- 
 press it in all equations is evidently absurd. Had we 
 always been accustomed to the new units instead of the 
 centimeter and second, there would have been the same 
 natural tendency to omit to express the velocity of light 
 and twice the Rydberg constant as there is now to omit 
 specific inductive capacity. 
 
 It is difficult to escape the conclusion that we are 
 one step nearer to a more complete understanding of 
 the real connection between matter and the aether of 
 space, that is to say, an understanding of the properties 
 of the aether itself. 
 
 It is not feasible to present this subject without a 
 limited use of mathematical symbols, which were pur- 
 posely avoided in my former book, "The Mystery of 
 Matter and Energy." The chief purpose in view in 
 that work was a statement of the aims and purposes 
 that constitute a definition of the goal. Since its pub- 
 lication in 191 7 much substantial progress has been 
 made toward the attainment of the goal, and it is not 
 now necessary to change the views expressed therein. 
 The mathematical sections of this work are, however, of 
 the simplest kind which students who have followed the 
 common undergraduate courses in the colleges may 
 read. Much may be obtained from the work without 
 following the mathematical processes at all. 
 
 June 14, 1919. 
 
NOTE 
 
 Since writing the above preface the attention of the 
 scientific world has been focused upon the recently an- 
 nounced results obtained during the total eclipse of the 
 sun, May 29 of this year, a report of which has just 
 been made to The Royal Society of London. These 
 announced results support the so-called "Relativity 
 Theory of Gravitation" due to Professor Einstein. Lest 
 there may some confusion exist in the minds of those not 
 familiar with Professor Einstein's theory because of the 
 name which has been applied to it, some remarks upon 
 this subject seem to be required, because the subject of 
 gravitation is discussed within these pages. 
 
 In a report to The Physical Society of London on the 
 "Relativity Theory of Gravitation," Professor A. S. 
 Eddington has summed up the matter on the last page 
 (91) in the following words, "In this discussion of the 
 law of gravitation, we have not sought, and we have not 
 reached, any ultimate explanation of its cause. A certain 
 connection between the gravitational field and the meas- 
 urement of space has been postulated, but this throws 
 light rather on the nature of our measurements than on 
 gravitation itself. • The relativity theory is indifferent 
 to hypotheses as to the nature of gravitation, just as it 
 is indifferent to hypotheses as to matter and light." 
 
 The recent result from the eclipse may be regarded as 
 one of the first, if not the first, experimental proof of the 
 truth of the theory of relativity, but it seems to the 
 author to be a misnomer to call the Einstein theory a 
 theory of gravitation, because it deals with one phase 
 only of a much larger general theory, which must assign 
 a cause for the gravitational force. The Einstein theory 
 
VIII 
 
 Note 
 
 admittedly assigns no cause for the force, and does not 
 connect it with the atoms of matter or with the motion 
 of the electrons within these atoms. The theory de- 
 veloped within these pages does connect the gravita- 
 tional force directly with the motion of the electrons 
 within the atoms. To obtain these results it is pointed 
 out that the theory of relativity is required, for this 
 theory is involved in the recent modification of electro- 
 magnetic theory due to Mega Nad Saha, who makes use 
 of the four-dimensional space of Minkowski and the 
 relativity theory. The common form of electromagnetic 
 equations is required to be modified to obtain the results 
 described in these pages, and the establishment of the 
 relativity theory strengthens the argument for the au- 
 thor's theory of gravitation. 
 
 There is no conflict between the theory of Professor 
 Einstein and that here given. On the contrary, they 
 supplement each other, both depending upon the theory 
 of relativity. The reason for adding these remarks is 
 the thought that it may naturally occur to any one that 
 there is room for but one theory of gravitation. So 
 there is, but such a theory must go to the root of the 
 matter and assign a cause for the gravitational force, 
 and the Einstein theory does not claim to do this, but 
 deals with a single phase of a broader comprehensive 
 theory. It is hoped that this explanation will have the 
 desired effect of removing any misunderstanding because 
 of a supposed conflict of theories. 
 
 The delay In the publication of this work because of a 
 printer's strike has afforded an opportunity to refer to 
 these very recent results obtained from the eclipse of 
 May 29 before the book goes to press. 
 
 Albert C. Crehore 
 November 19, 1919. 
 
Contents 
 
 Introductory. A brief statement of some of the results which 
 
 are described more fully in the following sections 1-20 
 
 II 
 
 AH matter divided into two classes, according to whether or 
 not it radiates energy. Matter in the second class, 
 while radiating energy, is treated by the author for the 
 first time. Previous atomic theories briefly reviewed. 
 The importance of the Lorentz mass formula. The 
 J. J. Thomson atom. The Rutherford atom. The 
 Bohr atomic theory, and the chief difficulties in ac- 
 cepting it. Brief statements as to the author's theory 
 more fully treated in subsequent sections 21-29 
 
 III 
 
 Application of electromagnetic theory to the problem. 
 Proof of the theorem that the mechanical force acting 
 upon a stationary atomic nucleus at a distance is pro- 
 portional to the sum of the vector accelerations of all 
 the electrons in the atoms of a gas resolved in a plane 
 perpendicular to the line joining the nucleus and the 
 center of the orbit of the electron 30-37 
 
 IV 
 
 Consequences of the above theorem. Theory agrees with 
 observation in showing that there is no apparent radia- 
 tion of energy in the normal state of a gas. Applica- 
 tion to hydrogen, the atom of which has two electrons 
 instead of a single one. Disturbed state of hydrogen 
 
Contents 
 
 gas when absorbing and radiating energy. The spectrum 
 of hydrogen, observed, and extended fay a mathematical 
 formula. Series of spectral lines known as the Lyman, 
 Balmer, and Paschen series. The X-ray spectrum of 
 hydrogen. A formula for the hydrogen spectrum equiv- 
 alent to the above. The new formula preferred because 
 it divides the hydrogen spectrum into series different 
 from the Lyman, Balmer, Paschen, etc., series, being 
 more suitable for -this atomic theory. The new series 
 are all of the first lines of each of the former series, all of 
 the second lines, third lines, etc. These new series have 
 different characteristics from the former series. The 
 whole of one series of lines emitted as radiation fre- 
 quencies in one operation of the electrons in return- 
 ing to the original orbit, which are responsible for the 
 series of frequencies. The theoretical ionizating volt- 
 ages in hydrogen agree very exactly with recent ex- 
 perimental values. There is not sufficient experimental 
 data for an exact solution of the problem 38-47 
 
 Fundamentally new point in the theory is that a whole series 
 of spectral lines is emitted at once by the electrons in 
 returning by a species of spiral paths. The stable orbit 
 in hydrogen has but one uniform size, and there are not 
 a multiplicity of stable orbits. Justification for making 
 certain definite assumptions. Assumption of the ex- 
 pression for the sum of the accelerations of the two 
 electrons in hydrogen (25), an infinite series. Deriva- 
 tion from this of the sum of the velocities (26), and the 
 sum of the position vectors (27) of the two electrons. 
 Assumption of expression for the difi'erence of the ac- 
 celerations (29). Derivation of the difi'erence of the 
 velocities (30), and the difi'erence of the position vectors 
 (31). From these are derived expressions for the in- 
 dividual accelerations (32), velocities (33), and position 
 vectors (34). Their values when the time is taken as 
 zero and infinity (35)-(52) 48-57 
 
Contents xx 
 
 VI 
 
 Determination of the constants in these equations by the 
 initial and final conditions of the motion. Frequency in 
 the final orbit twice the Rydberg constant. One con- 
 dition is that, when the time is zero, one electron has 
 reached its maximum distance from the nucleus and is 
 about to return. Its velocity must then be perpen- 
 dicular to its radius vector. The constants are shown 
 to depend upon the sums of infinite series. They are 
 only constant during one return of the electrons after 
 one excursion. On another occasion the constants may 
 have a different value and the paths be different. The 
 initial velocity, and therefore the kinetic energy, of each 
 electron the same as the final values 58-61 
 
 VII 
 
 Determination of the numerical values of the sums of the 
 infinite series that enter into the above equations and 
 their constants 62-69 
 
 VIII 
 
 Initial values of the position vectors of the two electrons in 
 
 hydrogen for all values of T2 70-73 
 
 IX 
 
 Application of the principle of the conservation of energy to 
 the system. The system not a so-called conservative 
 system. Application of the Einstein equation gives an 
 expression for the total energy radiated (108), which 
 differs for each series. The energy is a function of the 
 initial positions of the electrons. Determination of the 
 initial and final potential energies of the system, Vo and 
 V^. The latter is constant for all series since the final 
 radius is constant. The former, Vo, diff"ers from the 
 energy radiated by a constant, and is variable according 
 to the value of an integer, T2. The energy required to 
 separate electrons from the hydrogen atom is given by 
 the values of Vo. This energy converted into volts gives 
 
xii Contents 
 
 a table of ionizing voltages that agree with experimental 
 values in a remarkable manner. The minimum ionizing 
 voltage in hydrogen is 11.132 volts, which agrees closely 
 with observation. The maximum is 15.5 volts, which 
 is very near to 15.8 recently observed. The Bohr theory 
 gives a maximum of 13.54 volts, which is too low 74-81 
 
 X 
 
 Absolute value of the velocity, radius and kinetic energy of 
 the two electrons in the hydrogen atom determined. 
 Some numerical coincidences in the Bohr value of the 
 Rydberg constant. No theoretical support for this 
 value in the author's theory. New value for Rydberg's 
 constant found. It is connected with the Lorentz mass 
 formula. Numerical value of e'/mg obtained from the 
 expression for the Rydberg constant. Dimensions of 
 the Rydberg constant considered. The specific in- 
 ductive capacity, k, has the dimensions of the reciprocal 
 of a velocity. Combining the new expression for the 
 Rydberg constant with the well-known expression for 
 the electrochemical equivalent, absolute values of e and 
 mff are obtained, namely e = 4.763 x 10-'°, and ma = 
 1.658 X 10"**. The use of the Bucherer value of e/cmo 
 with the above gives> an absolute value of the mass of 
 the electron, mo = .898 X lo"" 82-89 
 
 XI 
 
 Value for the velocity of electrons in rings of electrons. Dif- 
 ficulty connected with the radiation of energy from 
 rings of electrons in electromagnetic theory discussed. 
 Absolute value of the radius of the orbit of electrons in 
 hydrogen discussed 90-gSi 
 
 XII 
 
 Computation of the orbits of the two electrons in hydrogen 
 in the caseofthe" head " series of spectral lines. Method 
 adopted is to expand e""' sin vt and eT>" cos n in series 
 in terms of powers of vt. pi + pi first calculated. Re- 
 
Contents xiii 
 
 suit shown in curve I, Fig. 5. pi next computed, and 
 shown as curve II, Fig. 5. From these p2 and pi — pi 
 are obtained graphically, and shown as curves III and 
 IV, Fig. 5. Curves II and III, pi and pa, in this figure 
 are the paths of the two electrons as they approach their 
 final orbit, the smaller circle in the figure. Labor in- 
 volved in these computations has limited the author 
 to a single example only, — that of the "head" series 
 of spectral lines 96-109 
 
 XIII 
 
 Atoms in the steady state not radiating energy. Funda- 
 mental problem to obtain an expression for the mechani- 
 cal force that one electron moving in a circle exerts upon 
 another in a different circle. Author has solved this 
 problem according to two of the forms of electromagnetic 
 theory, the J. J. Thomson equations, and the Lorentz 
 equations. Electromagnetic theory in process of de- 
 velopment. Valuable contribution by Maga Nad Saha, 
 who treats the problem by means of Minkowski's four 
 coordinate space. Modifications of current theory 
 pointed out by Saha important. Author's result, by use 
 of Lorentz equations only, fully described. Theory de- 
 mands the existence of forces at great distances from 
 bodies that vary according to the inverse square law. 
 Theory leads to a result not in harmony with the Law of 
 Equal Action and Reaction. The Doppler factor A 
 occurs as i/A' in the coefficient of the equation. Author 
 criticized by G. A. Schott for assuming this factor sensibly 
 equal to unity in taking the time average of the force. 
 Schott checks the author's result if this factor may be 
 assumed to be unity. He obtains a different result, but 
 still the inverse square law, when the Doppler factor is 
 
 1 -~ Schott's criticism met. Current electro- 
 
 cR 
 magnetic theory shown to require some modification. 
 Author obtains an equation that represents the gravi- 
 tational law in all respects. Dimensions considered. 
 Equation required to be multiplied by some quantity 
 having dimensions in terms of length and time in order 
 
xiv Contents 
 
 to make the two members agree. The mass of the 
 electron numerically satisfies this requirement. It is 
 probable that it also satisfies the dimensional require- 
 ment. This gives the dimensions of mass as the re- 
 ciprocal of specific inductive capacity, and, therefore, as 
 a velocity. Having found the dimensions of specific 
 inductive capacity, magnetic permeability and mass in 
 terms of length and time, a new table of dimensions of 
 units has been constructed in terms of length and time 
 only.. This table shown to be rational 1 10-121 
 
 XIV 
 
 The work of Saha shows that the dr in the Doppler factor 
 should not depend upon time alone, but equally upon 
 the four coSrdinates of the Minkowski generalized 
 space. Attraction between two material bodies. Space 
 average also required for the orientation of the orbits in 
 all possible ways. Derivation of a new expression for 
 the gravitational constant (213), simpler than that 
 heretofore published. It depends upon three quantities 
 only, e, mo and b. Dimensions correct. Former pub- 
 lished value included also ir*, c* and miA It is believed 
 that the Bohr expression for the Rydberg constant does 
 not represent a true relation between physical quantities. 
 Simple expression for the mass of a body (219). Dimen- 
 sions the same as those of the mass of the electron, mo . 122-131 
 
 XV 
 
 Applications of the gravitational equation. First, to find the 
 weights of rings of electrons on the earth's surface (221). 
 Attempt to find the combinations of rings of electrons 
 that make up different kinds of atoms on the basis of 
 the weights of rings. Table of combinations (223). 
 Example of magnesium. Table indicates a great pre- 
 ponderance of rings of four electrons (225). This may 
 be tested. Average velocity of an electron in the earth 
 found to be |8= .0071. Table of velocities of electrons in 
 rings (233). Velocity in ring of four, ;8 = .00728, very 
 close to the average for the earth. Result is general, ap- 
 
Contents xv 
 
 plying to any bodies, the earth, planets, sun, etc. Ex- 
 ample of a hydrogen star and a helium star. Avogadro 
 constant determined (244). The number of units of 
 electrical charge on the atomic nuclei not required to be 
 equal to the atomic number 132-146 
 
 APPENDIX A 
 
 The gravitational equation for two revolving electrons is 
 averaged for the orien tation of their two axes with re- 
 spect to each other. If there are four electrons in a 
 group having directions of axes parallel respectively to 
 the four medial lines of a regular tetrahedron, the average 
 gravitational attraction is independent of the orientation 
 of this tetrahedron. The average attraction of one 
 electron in this group is the same as that obtained above. 
 A system of such groups of electrons probably makes up 
 the structure of crystals 147-152 
 
 APPENDIX B 
 
 Proof that the number of electrons in one gram of substances 
 in general, hydrogen excepted, is a constant quantity, if 
 we start with the assumption that the number of elec- 
 trons per atom is proportional either to the atomic num- 
 ber or to the atomic weight 153-154 
 
 APPENDIX C 
 
 New expression for Planck's constant, b, found. It agrees in 
 dimensions with the new space-time system of dimen- 
 sions. The numerical value of Planck's constant ob- 
 tained from it is 6.558 X io~". This is very close to 
 the best result of Millikan's machine-shop-in-vacuo ex- 
 periment on the emission of electrons from fresh metallic 
 surfaces, namely 6.56 x io~". An equivalent expression 
 for b, involving only Rydberg's constant, the velocity of 
 light, and specific inductive capacity. A second equiva- 
 lent expression, involving the charge on one electron, 
 the velocity of light, the mass of the hydrogen atom, and 
 the specific inductive capacity. New units of length 
 
xvi Contents 
 
 and of time considered. The unit of time is the time 
 of one revolution of the electrons in hydrogen. The unit 
 of length is the distance traveled by light in this unit of 
 time. Velocity of light unity on this system. Twice 
 the Rydberg constant unity on this system. Specific in- 
 ductive capacity numerically equal to the velocity of light 
 in the C. G. S system, namely, 3 x 10". The mass of 
 the hydrogen atom on this system of units numerically 
 equal to the square of the unit charge, but it does not 
 have the same dimensions. It is also numerically equal 
 to the energy content of the hydrogen nucleus, but 
 does not have the same dimensions. The new unit of 
 mass equal to one gram times 3 x lo"", being approxi- 
 mately equal to a cube of water 31 meters on each edge. 
 The new unit of energy very large, being one erg times 
 (3 X 10'°)'. The energy contained in all the nuclei of 
 one gram of hydrogen sufficient to furnish energy at the 
 rate of one kilowatt for 2870 years. On the new 
 system of units Planck's constant takes a very simple 
 numerical form, h = (16/15&)*, where k is the specific 
 inductive capacity numerically equal to 3 x lo", but 
 the expression is not dimensionally correct without the 
 other factors which are unity numerically. This case 
 much like the common practice of omitting specific in- 
 ductive capacity in the C.G.S. system because it is 
 unity 155-161 
 
THE ATOM 
 
 I 
 
 HE question how best to present the new 
 theories contained in this volume has given 
 the author some anxious moments. It has 
 been customary among physicists to publish 
 papers representing original contributions to physics in 
 the best technical journals and the Proceedings and 
 Transactions of the Learned Societies, and the author has 
 followed this custom for many years. In the present 
 instance, however, the space required for a proper presen- 
 tation of the subject is so great that it would of necessity 
 extend the publication over a period of several months in 
 a series of articles in the standard journals. The alterna- 
 tive course of condensing the work into less space for this 
 purpose has been considered and rejected; for it is be- 
 lieved that in this instance the force of the argument 
 would be weakened, not strengthened, by abbreviation. 
 It is not contended that this is true in general, but it 
 seems to be particularly true of the present account. 
 There are many phases to the questions dealt with, and 
 all of them play a very definite part in assisting any one 
 to form a comprehensive idea of and competent judg- 
 ment of the whole. 
 
 It would be false modesty to try to disguise the fact 
 that the subject of atomic theory is treated in a new 
 form from the beginning to the end of the volume, and I 
 
The Atom 
 
 have endeavored to make clear the reasons why a new 
 treatment seems imperative. This has made it neces- 
 sary to emphasize the points where the current theory 
 of the atom, that due to Dr. N. Bohr, is deficient. 
 And this is the more necessary because this theory has 
 made a very strong appeal to physicists, who with some 
 reservations may be said to have adopted it as their 
 guiding theory. To do this will be regarded by those 
 who have their faces set only towards progress as no dis- 
 paragement of the work of Dr. Bohr. Indeed, the 
 author takes this opportunity to say that he regards the 
 work of Bohr as most suggestive, and as marking an 
 epoch in the progress of atomic theory. 
 
 It will not be beside the point to state briefly the 
 principal objections to the Bohr theory. It is held by 
 this theory that a uniform and constant frequency of 
 vibration of something takes place while an electron is 
 changing over from one circular orbit to another circular 
 orbit of smaller radius about the nucleus of the atom. 
 What it is that vibrates has never been shown and can- 
 not even be imagined. The path of the electron in chang- 
 ing over cannot possibly be a simple circular path, and 
 it is impossible to attribute the uniform vibration 
 supposed to exist to the form of the path of the 
 electron. 
 
 The new theory leaves us in no doubt on this matter, 
 and attributes the vibrations emitted during the radia- 
 tion from a gas directly to the forms of the paths followed 
 by the electrons themselves in returning to their normal 
 orbit after displacement. 
 
 This view of the matter does not require an infinite 
 number of possible stable orbits which are postulated in 
 the Bohr theory, but, in hydrogen at least, assumes that 
 there is but one fixed orbit in the normal state when not 
 
The Atom 
 
 radiating energy. When the electrons receive energy 
 from without, one of them is driven out to some maxi- 
 mum distance from the nucleus, depending upon the 
 amount of energy received, and returns immediately 
 thereafter to its former normal orbit, provided no second 
 pulse of energy is received in the meantime. The radia- 
 tion of the energy, which has been received, takes place as 
 it is returning to the original orbit, and the paths followed 
 by the electrons are responsible for the frequencies of the 
 vibrations observed in the spectra. 
 
 It has been attempted to work out the forms of these 
 paths that are possible paths such as will emit only the 
 radiation frequencies in the observed spectra, and, as a 
 guide for this purpose, use is made of a theorem which 
 has been estabhshed by means of the current form of 
 electromagnetic theory. Reference to this can scarcely 
 be made in these introductory remarks. 
 
 The physical processes attributed to the electrons are, 
 therefore, very different in the Bohr theory and in the 
 author's theory. In the former one operation of an 
 electron in changing over from one orbit to another is 
 supposed to emit but one single vibration frequency, 
 while in the latter an infinite series of frequencies is 
 emitted by the electrons in one operation, that of return- 
 ing toward the nucleus again after one excursion. 
 
 By applying the principle of the conservation of energy, 
 and by the use of the Einstein equation, which makes 
 the energy radiated equal to Planck's constant times the 
 frequency of vibration, it has been possible to obtain the 
 total energy radiated from the system by summing up 
 the terms of an infinite series, since an infinite series of 
 frequencies is emitted in one return of the electrons from 
 a single excursion. A simple expression for the energy 
 required to separate the electrons from the nucleus, if 
 
The Atom 
 
 this energy happens to be received while the electron is 
 at its maximum distance, has been obtained. From this 
 energy the ionizing voltages required for hydrogen have 
 been obtained, and they have been compared with the 
 recent experimental observations on hydrogen. The 
 agreement between these theoretical and experimental 
 values is remarkable and affords strong support for the 
 theory. The minimum ionizing voltage obtained from 
 the theory is 11.132 volts, and the maximum 15.5 volts. 
 Experimentally nothing is observed to happen in hydro- 
 gen until II volts is passed, and a recent experimental 
 result has discovered a new type of ionization in hydro- 
 gen at about 15.8 volts, which is in agreement with the 
 upper limit given by the theory. The Bohr theory gives 
 no indication of any ionizing voltage above 13.54 volts, 
 which is too low. Reference must be made to the text 
 for a table of the other ionizing voltages between these 
 two outside figures. 
 
 The new theory leads to the belief that the Rydberg 
 constant, which occurs in the formulae of the spectra of 
 every element whose spectra have as yet been reduced to 
 formulae, is connected very closely with the properties 
 of the atomic nucleus. A new expression for the Rydberg 
 constant, different from and simpler than that given by 
 Bohr, has been obtained from the well-known Lorentz 
 mass formula for an electrical charge at slow velocities, 
 
 4 e' 
 Sc'a 
 
 The new value for the Rydberg constant, K, is 
 
 If we solve the Lorentz mass formula for the radius. 
 
The Atom 
 
 a, and apply it to the nucleus of the hydrogen atom hav- 
 ing a mass mg, we have 
 
 i6 e2 
 
 in which it is seen that the reciprocal of. the expression 
 
 ttibI - ] involved in the Rydberg constant occurs. 
 
 Numerical values of the charge on the single electron, 
 the mass of the hydrogen atom, and the mass of the 
 electron have been obtained simply by the use of this 
 expression for the Rydberg constant coupled with the 
 well-known expression for the Faraday constant, in- 
 volving the electrochemical equivalent of an element in 
 electrolysis, and the Bucherer constant ratio of e to cmo 
 as follows, 
 
 e = 4.763 X 10"^" electrostatic units, 
 mn = 1.658 X io~^* grams, 
 mo = .898 X io~^' grams. 
 
 The only three experimental constants which have 
 entered into the determination of these two values are the 
 Rydberg constant, 3.290 X Iol^ the Faraday constant, 
 9649.4, and the Bucherer constant, 1.767 X I0^ each of 
 which constants are known with exceptional accuracy. 
 These results obtained from the new formula for the 
 Rydberg constant above given are within the limits of 
 accuracy of the independent experimental determina- 
 tion of e, niH and of mo, and there are strong reasons for 
 believing that the expression for the Rydberg constant 
 is a true relation between the quantities involved, and 
 that these are the correct values of e, mn, and mo. The 
 values obtained by Milhkan are e = 4.774 X lo-^"; 
 lUH = 1.662 X 10-"^ 
 
The Atom 
 
 The one thing that has held physicists to the Bohr 
 theory and has helped the theory more than anything 
 else, is the fact that a theoretical expression has been ob- 
 tained for the Rydberg constant as follows, 
 
 „ 27r^moe* 
 
 When numerical values of these constants are sub- 
 stituted in this expression, the agreement with the Ryd- 
 berg constant is surprisingly close to the third significant 
 figure. There seems to be no support in the new theory 
 for this expression for the Rydberg constant, and it is 
 beheved that the expression does not represent a true 
 physical equation. The reasons for holding this view 
 will presently be given, but first it seems worth while 
 to point out that thelre exist two other numerical co- 
 incidences in this expression which are just as close as 
 the agreement with the Rydberg constant. The sig- 
 nificant figures in the Rydberg constant are very close 
 indeed to the significant figures in ttVs = 3.289,868. 
 The usual value of the Rydberg constant is 3.290 x 10**, 
 which differs by only 1.3 in the fifth significant figure. 
 It must be, therefore, that the quantity 6moeV6' is very 
 close indeed to lo'^ an even number, for the Rydberg 
 constant may be written 
 
 TT^ 6moe* 
 
 K 
 
 h^ 
 
 Substituting the values of the constants as given by 
 MilHkan in the second factor, we find that it equals 
 0.999,53 X 10^*, a value very close to 10'*- 
 
 It seems to the author to be very unfortunate that the 
 Lorentz form of equations in electromagnetic theory 
 has been developed from fundamental equations that 
 
The Atom 
 
 omit to express the specific inductive capacity of the 
 medium, and by impHcation at least leaves one to under- 
 stand that it is of little consequence as being unity in 
 value and devoid of dimensions in terms of length and 
 time. That this is the common practice there is little 
 doubt. A reference to the work of Schott on Electro- 
 magnetic Radiation, which may be regarded as a repre- 
 sentative treatise on this subject, shows that the specific 
 inductive capacity, k, is ordinarily not expressed in any 
 of the fundamental equations of electromagnetic theory, 
 and indeed if there is any reference to it in this work it 
 has escaped the author's search. This electromagnetic 
 theory has led to expressions for quantities that are 
 equated to each other, and yet their dimensions are different 
 unless we are prepared to admit that k, the specific in- 
 ductive capacity, is dimensionless in terms of length and 
 time. The Lorentz mass formula above given may be 
 cited as a first example of the meaning. In the form 
 expressing the radius of the atomic nucleus above, the 
 dimensions of the left side of the equation are simply 
 those of length, L. On the electrostatic system of units 
 the dimensions of quantities are expressed in terms of 
 the four fundamental quantities L, M, T and k, and the 
 dimensions of the expression on the right of the equation 
 are Lk~^, and not simply L, as in the left member. If 
 k is dimensionless in terms of L and T, then it becomes a 
 true physical equation, but if it is not, then it leads to 
 erroneous results to write these equations without the k. 
 It is believed that the true dimensions of k, the specific 
 inductive capacity, are those of the reciprocal of a velocity, 
 namely L~^T, and we are not prepared to admit that k 
 is dimensionless in terms of L and T. If this is so, then 
 the k must always be expressed and not omitted in order 
 
8 The Atom 
 
 to make true physical equations, and we will have to 
 write the Lorentz mass formula 
 
 Now, the reciprocal of the quantities on the right side 
 of this equation occur in the expression for the Rydberg 
 constant given above. The dimensions of the Rydberg 
 constant are simply those of a frequency, or the recipro- 
 cal of a time, namely T~^. Hence, the dimensions of the 
 quantities on the right of the above equation should be 
 those of the reciprocal of the Rydberg constant, namely 
 simply a time, T. By giving to k the dimensions of the 
 reciprocal of a velocity, Lr^T, the left member, ok, does 
 become simply a time, T, thus agreeing with the Rydberg 
 constant. On the electrostatic system of units the 
 dimensions of the expression we have given above for 
 the Rydberg constant, namely 
 
 2K = mE('^^^ 
 
 are Lr^hr^. If, now, k has the dimensions of the 
 reciprocal of a velocity, Lr^T, then these dimensions 
 are simply T-^, and the expression represents the Ryd- 
 berg constant not only in magnitude but in dimensions. 
 Now the equation connecting k and ix, the specific in- 
 ductive capacity and the magnetic permeability, was 
 first pointed out by Maxwell, and the dimensions of the 
 product of k and jn have been known for many years 
 from this relation, namely 
 
 where c is the velocity of light. The dimensions of the 
 product are, therefore, Lr^T^. It is almost proved by 
 
The Atom 
 
 this that both k and /x have some dimensions in terms of 
 L and T. It is most improbable that all of the dimen- 
 sions of the product fall upon /x alone, and that k is 
 dimensionless. Also, a determination of the dimensions 
 of one of the two quantities automatically determines 
 the other because of this equation. If k has the 
 dimensions of the reciprocal of a velocity, then fx 
 receives the same dimensions, and they each thus repre- 
 sent the same kind of quantity. There has been some 
 speculation in the past as to the dimensions of the ratio 
 of k to fj,, for, if this could have been determined, of course 
 each might be found. It is interesting to observe that 
 the values above found make the ratio have no dimensions 
 in terms of L and T, being in this respect like the quantity 
 /3, the ratio of two velocities, a pure numeric. 
 
 When we examine the expression for the Rydberg 
 constant as given by Bohr, namely 
 
 we find that the dimensions of the left member of the 
 equation are those of the Rydberg constant, T~\ and 
 the right member on the electrostatic system of units has 
 the dimensions 
 
 and on the electromagnetic system 
 
 If k is not dimensionless, it ought to be expressed in 
 the above equation. Possibly it is understood that it 
 ought to be there, but this practice of omitting to write 
 it down seems to the author to be pernicious, because it 
 may easily lead to confusion. 
 
 There are two equally compelling reasons for believing 
 that the dimensions of mass are those of a velocity. 
 
10 The Atom 
 
 LT~^, the reciprocal of specific inductive capacity and 
 magnetic permeability. The first is to be found in a 
 new expression for Planck's constant, h, and the second 
 in the gravitational equation, which will be referred to 
 later. 
 
 The new expressions for Planck's constant, h, are given 
 in Appendix C, below. The three equivalent values are 
 
 (f)" 
 
 where ub is the radius of the hydrogen nucleus, K, Ryd- 
 berg's constant, and c the velocity of light, and again 
 
 {iskyKc' 
 
 where k is the specific inductive capacity, and finally 
 
 \15kJ msc' 
 
 The second and third expressions are derived from the 
 first directly by the use of the Lorentz mass formula and 
 the new expression for the Rydberg constant above 
 given. The second form makes the numerical value of 
 h depend only upon the numerical values of the Rydberg 
 constant, K, and the velocity of light, c, since the specific 
 inductive capacity, k, is numerically unity. Both of 
 these constants are known with exceptional accuracy. 
 The resulting numerical value, taking K = 3.290 X 10'* 
 and c = 3 X lo^", gives 
 
 h = 6.5579 X io~". 
 
 This is in almost exact agreement with Millikan's ex- 
 perimental value of b, obtained from his machine-shop- 
 in-vacuo apparatus designed for the purpose of testing 
 the validity of the Einstein equation, which makes 
 
The Atom II 
 
 energy equal to Planck's constant times a frequency. 
 The best value of h obtained by Millikan as the result of 
 all these experiments on the emission of electrons from 
 metals by light of different frequencies was 
 
 b = 6.56 X 10-". 
 
 If we had used a value of the velocity of light slightly 
 less than 3 X 10*", which would be more exact, we would 
 have obtained a slightly larger theoretical value of h, 
 and a value in almost exact agreement with Millikan's 
 experimental value, since it differs as given by about 
 two units in the fourth significant figure. There seems 
 to be some diff"erence of opinion among authorities as 
 to the exact value of the velocity of light, and, hence, 
 the even number, 3 X lo'", has been adhered to. 
 
 Millikan has published a value for h = 6.547 X io~^', 
 as representing the most probable value of h when all 
 sources are taken into account, has given some weight 
 apparently to the experimental results of others, and has 
 struck a mean value. So great a difference as that be- 
 tween 6.56 and 6.547 would make an easily perceptible 
 difference in the slope of the straight line from which he 
 derived his own experimental value. It is the author's 
 opinion that there was no gain in the accuracy of the value 
 of h by abandoning the best result of his. own experi- 
 ments and giving so much weight to the results of others. 
 
 The dimensions of h are those of energy divided by a 
 frequency, that is, multiplied by a time. Hence these 
 dimensions may be classed with mechanical units rather 
 than electrical and magnetic units, the distinction being 
 that the former do not involve specific inductive capacity 
 and magnetic permeability. The dimensions of energy 
 are DMT-\ and multiplying by a time, the dimensions 
 of ib must be UMT'K 
 
12 The Atom 
 
 The first two expressions for the value of h given 
 above leave no doubt as to the dimensions, for the di- 
 mensions of K, c and ag do not involve specific inductive 
 capacity or magnetic permeability in any way. The di- 
 mensions of these first two forms are clearly L'T~*- If 
 these dimensions are equated to those of h, we have 
 
 This can only be a true relation if the dimensions of 
 mass are LT-^, that is to say, those of a velocity. 
 
 The third expression shows that the dimensions of b 
 are the same as those of the square of the electrical charge 
 of the electron, e. For the mnc^ in the denominator has 
 the dimensions of the fourth power of a velocity, and the 
 k* is the reciprocal of this, thus making the whole de- 
 nominator dimensionless in terms of L and T. 
 
 As illustrating the importance of always expressing 
 the specific inductive capacity, some consideration is 
 given to the use of a new system of units of length and of 
 time instead of the present centimeter and the second. 
 The second is not a natural unit of time as applied to 
 atoms, depending as it does upon the rotation of the 
 earth, neither is the centimeter a natural unit of length. 
 The new units considered are the time of one revolution 
 of the electrons in the hydrogen atom, 1/2 K, and the dis- 
 tance traveled by light in this time. On this system of 
 units the velocity of light becomes unity instead of 
 3 X 10'", and twice the Rydberg constant becomes unity. 
 Specific inductive capacity has the numerical value 
 3 X 10^" instead of unity as in the C.G.S. system. The 
 importance of not omitting to express it is, therefore, 
 very evident. 
 
 The above expression for the Rydberg constant, namely 
 
The Atom 13 
 
 makes e^ not only numerically equal to the mass of the 
 hydrogen atom, since 2K and c are each unity, but the 
 third expression for b above, since e^ = mn numerically, 
 and c = I, makes h equal to 
 
 ( — 7 ] numerically 
 
 but not dimensionally. And, since k on this system is 
 equal to the velocity of light on the C.G.S. system, this 
 expression shows that the nupierical value of Planck's 
 constant is made to depend upon an accurate value of 
 the determination of the velocity of light only. The 
 numerical value of h thus obtained, taking .^ = 3 x 10^", is 
 
 h = 1.598 X 10-^2. 
 
 To convert this over into the C.G.S. system of units only 
 requires an accurate knowledge of the Rydberg constant. 
 This is the same result as shown in the second expression 
 for h above, which makes it depend only upon K and c. 
 
 Had we always used the units now under discussion 
 for length and for time, we would have been inclined to 
 omit the c and the 2K, as being unity, from these equa- 
 tions in a manner exactly analogous to the tendency to 
 omit the specific inductive capacity on the C.G.S system 
 because it is unity. This example has served not only 
 to make this apparent, but it has also pointed out some 
 new relations, that the mass of the hydrogen nucleus 
 is numerically equal to the square of the charge of the 
 electron, and again numerically equal to the energy 
 content of the nucleus itself, since we may regard ttiec^ 
 or iKe^ as this energy content, the dimensions of energy 
 being those of the cube of a velocity. Mass is not, how- 
 ever, dimensionally the same as energy, the latter being 
 the cube of the former. 
 
14 The Atom 
 
 In a later section of this work an expression has been 
 derived from the Lorentz electromagnetic equations for 
 the Newtonian constant of gravitation, which assumes 
 the very simple form 
 
 , _ I _^ 
 
 fi 4 1 
 
 the k here being a different k from specific inductive 
 capacity, namely the Newtonian constant. The Lorentz 
 equations lead to a form of force-equation which may be 
 represented as follows, 
 
 F = Ceieir~^, 
 
 in which C is some constant, simply a numeric without 
 dimensions. The dimensions of a force on the left side 
 of this equation are LMT~^, but the dimensions of the 
 right member on the electrostatic system are LMT~%, 
 which is not a force unless the k is dimensionless in terms 
 of L and T, which we are not prepared to admit. Now, 
 it has been found that, if the quantities on the right of 
 this equation are multiplied by the mass of the electron, 
 jTio, we obtain numerically a value which makes the 
 magnitude of the force equal to the force of gravitation, 
 and this has led to the obtaining of the above value for 
 the gravitational constant. As the equation stands, 
 the right member requires to be multiplied by some kind 
 of physical quantity that has dimensions in terms of L 
 and T in order to make the two members agree in di- 
 mensions. And, since we have found that the mass of 
 the electron satisfies the numerical requirements, it is 
 reasonable to suppose that it also satisfies the dimen- 
 sional requirements. After multiplying by a mass, the 
 dimensions of the right member become 
 
 LMT^Mk, 
 and the value agrees with a force numerically, which has 
 
The Atom 15 
 
 the dimensions LMT'^. It is natural to conclude that 
 Mk has zero dimensions in terms of L and T, and that 
 the dimensions of mass are the reciprocal of those of 
 specific inductive capacity. Taking the latter as the 
 reciprocal of a velocity, we may consider mass as a ve- 
 locity of something and give it dimensions in terms of 
 L and T, namely LT~^. We have thus been led to con- 
 clude that neither k nor mass are fundamental units, 
 but that each may be expressed in terms of L and T. 
 Using these values of k and M in terms of L and T, a 
 new table of dimensions has been constructed, which 
 may be called the space-time system of units. This is 
 given in (202I) in the text. 
 
 This table itself constitutes an argument in support of 
 the theories that have led to it. Quantities that have 
 already been suspected to be of exactly the same nature 
 receive the same dimensions in the new system of units. 
 For example, quantity of electricity has the same di- 
 mensions as quantity of magnetism; electromotive force 
 the same as magnetomotive force; the coefficients of self 
 and mutual induction the same dimensions as electrical 
 capacity; electric force the same as magnetic force. If 
 it may be assumed that we now possess a correct system 
 of the dimensions of units in terms of length and time 
 only, it will prove to be a powerful tool for the proper 
 examination of physical quantities. 
 
 A gravitational equation has been obtained (201) by 
 means of an application of electromagnetic theory as 
 applied to the normal state of atoms while not radiating 
 energy, which represents all the laws contained in New- 
 ton's statement, but which attributes the cause of the 
 force to the electrons themselves in their motion about 
 the nuclei of their respective atoms. This equation not 
 only gives the approximate magnitude of the gravita- 
 
1 6 The Atom 
 
 tional force and leads to the simple expression above 
 given for the Newtonian constant, but it shows that the 
 force is always an attraction obeying the inverse square 
 of the distance law, and that it is proportional to the 
 product of the masses of the two bodies, and it also shows 
 that the attraction is independent of the orientation of 
 the two bodies, whether they be crystals or any other 
 form of matter, — solid, liquid or gaseous. 
 
 For those who have followed the author's work through 
 the published articles in the physical journals, it seems 
 to be required to refer to a criticism of this particular 
 phase of the subject that has been made by G. A. Schott. 
 This matter will be found discussed in the text, and it is 
 believed that the criticism has been fully met. It is not 
 necessary to repeat the arguments here, but it may be 
 said that the point of the criticism centered upon the 
 question whether it was legitimate or not to assume that 
 the Doppler factor, 
 
 is sensibly equal to unity when the time average of the 
 force is taken as the electrons circulate about their re- 
 spective nuclei. On the assumption that it is sensibly 
 equal to unity, Schott has verified the author's con- 
 clusions. But, on the assumption that it is equal to the 
 expression just given, he has shown that the result is 
 different. In each case, however, the resulting force 
 obeys the inverse square of the distance law. The re- 
 sult of Schott does not represent the gravitational force 
 in any other respect, while the author's result does repre- 
 sent it in a very complete manner. The recent work of 
 Saha^ h as made it evident that the St in the Doppler 
 ' Mega Nad Saha, Pbil. Mag., Vol. 37, No. 220, April, 1919, p. 347. 
 Pbys. Rev., Vol. XIII, N. S., Jan. 1919, p. 34; March, 1919, p. 238. 
 
The Atom 17 
 
 factor should not refer to time only, but to each of the 
 four coordinates in a generahzed Minkowski space, and 
 there are strong grounds for the belief that this Doppler 
 factor will have to undergo a modification in any revised 
 new form of electromagnetic theory. If any change is 
 made in this, the work of Schott in using the value just 
 as it stands is of little value. It seems most probable 
 that a similar multiplying factor will have to come into 
 the second term of the expression for the Doppler factor, 
 if, indeed, it is to be called the Doppler factor any more, 
 of the same nature as the factor that the author has found 
 to be required for the whole force itself, namely that of 
 the mass of the electron. This modification will make 
 the factor sensibly equal to unity, and the results ob- 
 tained on the assumption that it is unity, which the 
 author made, are of considerable interest because they 
 result in an exact expression for the gravitational laws. 
 
 An application of the gravitational equation to gross 
 matter has led to the expression for the mass of a body 
 as follows, 
 
 The summation is to be extended to every electron in 
 every atom of the body. The dimensions of this expres- 
 sion are correct, for S/3^ is a numeric without dimensions, 
 and the ratio e^/h has the dimensions LT~% on the electro- 
 static system. Putting k as the reciprocal of a velocity, 
 this becomes dimensionless on the space-time system of 
 dimensions. Hence m has the dimensions of mo alone, 
 and represents a mass. 
 
 It is shown that this expression for the mass of a body, 
 which is derived primarily from its weight, is equivalent 
 to the expression obtained by summing up the total 
 number of nuclei of the atoms in the body, the mass 
 
1 8 The Atom 
 
 really residing in the nuclei. It is well known that the 
 weights of bodies are strictly proportional to their masses; 
 but the two physical concepts of mass and weight are 
 not the same and should be carefully distinguished. If 
 we could stop all of the electrons from coursing around 
 their orbits, the weight would vanish but the mass would 
 not, and, since we cannot change the velocities of the 
 electrons, the weights and masses remain proportional. 
 Or rather, if we could stop them, the atoms would dis- 
 integrate and cease to exist as atoms and the body would 
 be recognized no longer. 
 
 A further application of the gravitational equation to 
 find the weights of rings of electrons on the earth's sur- 
 face makes the weights of these rings depend chiefly 
 upon the number of electrons in the ring whether the 
 ring happens to be in one kind of an atom or any other 
 kind. This has led to the conception that it may be 
 possible to find the particular combination of rings of 
 electrons that exist in the various kinds of atoms, for the 
 sum of the weights of the rings must equal the weight of 
 the atom, and, knowing the weights of the rings, we may 
 find a combination that gives the proper weight of the 
 atom. A table (223) has been given of the combination 
 of rings thus found. The method is less uncertain when 
 applied to the elements of low atomic weight, but it has 
 been extended clear through the periodic table of the 
 elements, including uranium, largely because the scheme 
 of the combinations seems to be revealed by the elements 
 of low atomic weight. The scheme indicates that atoms 
 are made up of rings of four electrons in much greater 
 numbers than rings of any different number. For ex- 
 ample, in the table as given the total number of the rings 
 of four in the seventy elements is 1470, as compared 
 with the next largest number 185 rings of two electrons. 
 
The Atom 19 
 
 It is not contended that all of these combinations of 
 rings are correct and will never be subject to change, 
 but the main feature of the table, the great preponderance 
 of the numbers of rings of four, may be tested by means 
 of the gravitational equation. For, in any mixed mass 
 of matter, such as the earth for example, it is necessary 
 that the average speed of a single electron shall be very 
 close to the speed of an electron in a ring of just four 
 electrons. By writing down the gravitational equation for 
 the earth as one body and for a single hydrogen atom on 
 its surface for the other body, it happens that the only 
 unknown quantity in the resulting expression is the sum 
 of the squares of the speeds of all the electrons in the 
 earth, which quantity may, therefore, be found. When 
 this is divided by the total number of electrons in the 
 earth, which is also known because it is equal to the mass 
 of the earth in grams times the number of electrons per 
 gram, we thus find the average speed of a single electron 
 in the earth. The number of electrons per gram is known 
 to be approximately equal to the Avogadro constant. 
 The result of this calculation gives the average velocity 
 of an electron in the earth as .0071. The theoretical 
 velocities of electrons in rings are for a ring of two, .00364, 
 a ring of three, .00546, a ring of four, .00728, of five, .0091, 
 and a ring of six, .0109. All of these velocities are in 
 terms of the velocity of light as being unity. The agree- 
 ment of the velocity of the average electron in the earth 
 with the velocity of an electron in a ring of just four 
 electrons is very close. Rings of three and of two should 
 reduce this average somewhat below that of a ring of 
 four, and we see that .0071 is slightly less than .00728. 
 
 It is pointed out that the mass of the earth in grams 
 came into both expressions used in the above calculation, 
 namely in the sum of the squares of the velocities of all 
 
20 The Atom 
 
 the electrons in the earth and the number of electrons 
 in the earth, so that when we divided the one by the other 
 the mass of the earth in grams canceled out, and no error 
 was, therefore, introduced by any uncertainty in the 
 numerical value of the mass of the earth in grams. This 
 fact points to the conclusion that the result obtained is 
 very general and would be true of any body whatever, 
 the planets and the sun just as well. This is in later 
 sections shown to be true, and considering all of these 
 matters there are strong grounds for thinking that we have 
 established in a fairly positive manner the proposition 
 contained in the atomic weight table (223), that the 
 great majority of the total number of rings of electrons 
 in atoms. is just the ring of four electrons. 
 
II 
 
 N presenting new ideas on any subject it is 
 natural to draw a close comparison between 
 the new and the old, for the new conceptions 
 would not be required if existing theories 
 were entirely adequate and stood in complete harmony 
 with the experimental facts as we know them. The 
 places where the prevailing theory seems to be deficient 
 require to be pointed out, so that it shall appear by com- 
 parison how completely these difficulties are removed by 
 looking at the matter in a new way. Heretofore atten- 
 tion has been given to atoms when they are neither radi- 
 ating nor absorbing energy from without, which is often 
 referred to as their steady states, because in this condi- 
 tion it has been held that electromagnetic theory as 
 applied to moving charges of electricity ought to be ap- 
 plicable to the electrons in atoms while in this state. All 
 matter has thus been divided into two great classes, 
 according to whether its atoms are or are not radiating 
 energy. 
 
 In the subject immediately before us, however, at- 
 tention is given for the first time to the atoms in 
 their second state while absorbing and radiating energy, 
 and we shall begin with a brief review of some of the 
 conceptions prevailing to-day as to the structure of the 
 atom. The first theory of atomic structure which had 
 a definite character was offered by Sir J. J. Thomson at 
 a time not long after his discovery of the separate exist- 
 ence of the electron. In this he postulated that each 
 atom consisted of a positive sphere of electrification of 
 
22 The Atom 
 
 fairly large dimensions within which a number of elec- 
 trons having a negative charge were circulating in orbits. 
 His reason for assuming the existence of this positive 
 sphere of electrification must have been to provide the 
 means for retaining the negative electrons within the 
 atom, for by this assumption he secured equilibrium for 
 the negative electrons, which are supposed to repel each 
 other. 
 
 The electromagnetic theory as applied to an electron 
 itself by several independent investigators, and in par- 
 ticular by H. A. Lorentz, whose so-called "sohd electron'' 
 has received the most attention, has pointed out what 
 is known as a "mass formula" for the several forms 
 of electrons. The mass formula for the Lorentz electron 
 is as follows: 
 
 Transverse mass = - -5- (i — &),~^ . . . (i) 
 5c^a ^ ' 
 
 A, e^ 
 Longitudinal mass = - -j- (i - fi^)~i. ... (2) 
 
 These expressions tell us that the mass of the electron is a 
 function of its velocity, /3c, and that mass is a vector 
 quantity depending upon the direction being considered. 
 That is to say, it differs in different directions. How- 
 ever, when the velocity is small compared with that of 
 light, the mass approaches the constant value 
 
 4, e^ 
 wio = - -r> (3) 
 
 where e denotes the electrical charge, c the velocity of 
 light, and a the radius of the sphere of the electron. 
 For values of the velocity of the electron greater than say 
 one tenth of the velocity of light, this theory shows an 
 appreciable increase in the mass with increasing velocity, 
 the limit being an infinite mass when jS = i. 
 
The Atom 23 
 
 The experiments of Kaufmann in measuring the mass 
 of the electron for various values of its velocity up to 
 very high velocities showed a variation in the mass with 
 the velocity in fairly good agreement with the results 
 obtained from electromagnetic theory above mentioned. 
 
 This good agreement between electrical theory and 
 observation has led to the belief that all ordinary mass 
 has an electromagnetic character, and that there is but 
 one kind of mass. Granting that this may be a fact, 
 it becomes obvious by means of equation (3) that we may 
 find the radius of the Lorentz negative electron, a. It is 
 
 a = - -i — , (4) 
 
 or numerically 
 
 a = 2.25 X io~^'cm (5) 
 
 The chief reason for alluding to this well-known history 
 is that this mass formula (4) shows that the radius of 
 the electron is inversely proportional to the mass for slow 
 velocities. If, therefore, the mass of the positive nucleus 
 of the atom is many times greater than the mass of the 
 negative electron, as it is known to be, then its radius 
 should be proportionally smaller than that of the negative 
 electron instead of larger than it. 
 
 These ideas are contrary to the conception of an atom 
 as first suggested by Sir J. J. Thomson above mentioned, 
 who made the positive charge of the atom occupy a larger 
 volume than the electron many times over. This reason, 
 and the fact that it was proved by experiments on the 
 scattering of alpha particles that the central positive 
 charge of atoms must occupy a very small space indeed, 
 led Sir Ernest Rutherford to propose a new atomic theory. 
 In which the positive charge of the atom is supposed to 
 be of extremely small dimensions, the electrons not being 
 
24 The Atom 
 
 within the positive charge but outside of it, circulating 
 about it in orbits that are large in comparison with 
 either the nucleus or the electron itself. 
 
 This Rutherford theory was, however, beset with 
 theoretical difficulties from the beginning because it 
 could not be shown by the accepted form of electro- 
 magnetic theory that there existed any possible orbits 
 for a group of electrons which would be stable orbits. 
 And thus it has come about that men have come to be- 
 lieve in a form of atom that the accepted form of electro- 
 magnetic theory cannot sustain, and yet this very form 
 has been forced upon us by certain other applications of 
 electromagnetic theory as applied to the electrons them- 
 selves, which has been outhned above. It must be that 
 electromagnetic theory has certain good features; that 
 it represents a truth at some points at least, but that it 
 is deficient and unsatisfactory in other points, so that it 
 becomes a puzzle when it is safe to use it and when it is 
 unsafe. That some modification of electromagnetic 
 theory is possible, which will bring it into line with these 
 atomic phenomena, makes a very strong appeal to right 
 reason. 
 
 It was at the time that this Rutherford theory of the 
 atom was laboring under these difficulties that Dr. N. 
 Bohr came to the rescue by providing the means whereby 
 these unstable electrons might find stability. The rescue 
 was not, unfortunately, effected by any modification of 
 electromagnetic theory, which might have again given us 
 a sense of security as being based upon a solid foundation 
 that was understandable, but was secured by bringing 
 to bear upon the Rutherford atomic theory the ideas of 
 Planck and of Einstein, which admittedly are not founded 
 upon electromagnetic theory, but are chiefly based upon 
 experimental evidence. This evidence is so strong that it 
 
The Atom 25 
 
 is difficult to disbelieve in the truth of the assertions 
 contained in what is now known as Planck's quantum 
 theory, and indeed there is no desire to disbelieve in this 
 theory; for the only desires of the true investigator are 
 to learn the truth whether he yet understands the reason 
 for it or not. At the same time there exists a very strong 
 and natural desire that these truths shall some day be 
 shown to be the result of a more comprehensive and 
 general electromagnetic theory than we now possess. 
 We may say, therefore, that it now becomes one of the 
 chief and legitimate aims of the investigator to seek to 
 discover some modified form of theory which may har- 
 monize all of these various phenomena that now have no 
 interpretation in terms of the accepted form of it. 
 
 This very conclusion compels us to doubt the general 
 applicability of the present form of the theory in atomic 
 phenomena. At the same time the present theory gives 
 a good account of itself at certain points, as we have 
 pointed out above in one instance, namely by showing 
 that the positive nucleus of the atom should occupy a 
 very small volume. We are not justified in throwing it 
 all aside, for it has proved itself to be correct in too many 
 instances, and, moreover, without it we are entirely at 
 sea, and everything seems confused and without any 
 theoretical basis. We must use the theory in part and 
 learn to distinguish when possible between those cases 
 where it is applicable and where it is not. 
 
 Hereafter it cannot, therefore, be regarded as illegiti- 
 mate or even strange when certain assumptions are made 
 that are not in strict accord with the current form of 
 electromagnetic theory, for it is only by such attempts 
 that there is hope eventually of discovering some modi- 
 fication of the present theory that will harmonize 
 everything. 
 
26 The Atom 
 
 We shall content ourselves by giving very briefly a 
 statement of some of the chief features introduced by 
 Dr. Bohr into the theory of the Rutherford atom, by 
 which the solution of the question of atomic structure 
 assumed a definite form. Starting with the simplest 
 atom, that of hydrogen, he has concluded that in its 
 normal neutral condition there is a single nucleus of 
 charge plus e and a single electron of charge minus e 
 circulating around the nucleus in an orbit. When the 
 atom is neither absorbing nor radiating any energy this 
 orbit is supposed to have a circular form, but strangely 
 enough, the radius of it may have a theoretically infinite 
 number of values at different times, and the electron may 
 be stable in any one of this large series of different-sized 
 orbits. The actual velocity of the electron is supposed 
 to differ in each of these so-called stable orbits, and, 
 therefore, the kinetic energy of the electron differs in 
 every one of them. In each instance, however, the 
 attraction between the electron and the nucleus is sup- 
 posed to obey the inverse square of the distance law, 
 and the velocity in each orbit is so adjusted that the 
 centrifugal force of the electron due to its mass is exactly 
 balanced by the force of attraction toward the nucleus. 
 
 Next, as to the manner in which such an atom emits 
 its energy in the form of vibrations that correspond ex- 
 actly with the vibrations that hydrogen atoms are known 
 to emit by observations of the spectrum of hydrogen. 
 If such an atom should receive energy from some ex- 
 ternal source, it is supposed that the electron suddenly 
 goes outward from the nucleus, moving from the orbit 
 in which it then happens to be to some one of the other 
 larger orbits, which one depending upon how much 
 energy has been received. After all the energy has been 
 received that is coming to it on this one occasion it is 
 
The Atom 27 
 
 then free to radiate energy again by dropping back to 
 some smaller orbit. There is a difficulty here in seeing 
 why it should leave the larger orbit at all if it is in equi- 
 librium there, after it has been brought to it by the receipt 
 of energy. And again, it is difficult to see in which one 
 of the various possible stable orbits between its outer- 
 most position and the smallest possible orbit it will stop 
 in its course back toward the nucleus. It is true that it 
 will radiate the most energy if it goes clear through to the 
 last or smallest orbit, which is considered the most stable 
 position; but, if it does this every time it is displaced, 
 then the size of the orbits in a normal mass of hydrogen 
 gas, while not radiating energy, must all be alike and 
 equal to the smallest possible orbit. If this is the case, 
 there seems to be no possible utility in postulating a 
 large number of stable orbits, if the electron is never to 
 stay in any one of them. For the return to the nucleus 
 must begin immediately after it has reached its maxi- 
 mum distance, and there would be no time in which it 
 remains stable in its larger orbit. If, on the other hand, 
 it does remain in its larger orbit after receiving the initial 
 energy that drove it there, and waits there until the next 
 disturbance is received, it is true that this disturbance 
 might subtract energy instead of add it, and thus assist 
 in bringing it back to any smaller orbit. Admitting this 
 to be the case, it also follows that the new impulse might 
 drive it further away, and hence it is probable, on the 
 theory of chance, that one first impulse may drive it 
 from the first to the second orbit, and that it will remain 
 there for a time; then a second impulse might follow 
 that drives it to the third orbit, and this again be fol- 
 lowed by one which drives it one further on, and so forth. 
 The series of successive impulses thus at the last drives 
 it completely away from the nucleus ionizing the gas. 
 
28 The Atom 
 
 The ionizing voltage required to accomplish this on the 
 part of the bombarding electrons will, therefore, be no 
 greater than the greatest of the links in this chain of 
 happenings. The numerical value of the ionizing voltage 
 given by this hypothesis, however, does not agree with 
 the results of experiments on hydrogen. There is scant 
 experimental support for the assumption of a large series 
 of stable orbits. 
 
 Now as to the frequencies of vibration at which this 
 radiated energy is dissipated. When the electron changes 
 over from one of these larger orbits to a smaller one it 
 also changes from one frequency of revolution to another 
 around the nucleus. These frequencies of revolution are 
 such according to the theory that the difference between 
 the frequencies of revolution of the electron in any two 
 of the orbits whatever is strictly proportional to some 
 one of the frequencies which are known to be emitted 
 by hydrogen as found in the observations of the spectrum 
 of hydrogen. The theory assumes, therefore, that but 
 one single harmonic frequency is emitted each time an 
 electron changes over from a larger to a smaller orbit. 
 That is to say it is assumed that there exists a uniform 
 harmonic vibration of something during the time that the 
 electron is changing orbits. Nothing is said as to the 
 path by which the electron makes the change. It is 
 impossible that the path should be a simple circulas 
 path, for no circular arc can be drawn connecting the 
 outer orbit with the inner one without some sudden 
 transition In the direction of motion, and we have the 
 hypothesis inherent in the theory that the electron Is 
 emitting a perfectly uniform and fixed frequency while 
 it moves in some form of path that cannot be simple. 
 
 When this, the greatest and most Important objection 
 to the Bohr theory, is pointed out to an advocate of this 
 
The Atom 29 
 
 theory, I have heard the reply that this is " the assump- 
 tion." It is admitted that we do eventually come to 
 some last thing in the course of explaining things in 
 terms of other things, which itself cannot be explained 
 in terms of anything. There should, however, be some 
 choice allowed in selecting our last trench for a final 
 stand. It must stand certain tests and be at least reason- 
 able. If something less objectionable can be substituted 
 for the above almost unthinkable assumption of a uni- 
 form vibration emitted during a complex motion of an 
 electron, it may perhaps get a hearing. 
 
 In the theory about to be described we are left in no 
 uncertainty as to the source of the vibrations which 
 cause the lines in the spectrum of hydrogen. They are 
 due to the very motion of the electrons themselves in 
 returning to the nucleus again after displacement by the 
 receipt of energy from the external source. The forms of 
 the paths followed by the electrons may emit a whole 
 series of frequencies at once, all of which are, however, 
 to be found in the observed spectrum. But this state- 
 ment anticipates the logical presentation of the theory, 
 which is based chiefly upon an application of the electro- 
 magnetic theory to the problem. 
 
Ill 
 
 (OME light is thrown upon the subject by a 
 consideration of the mechanical force that 
 one single moving electrical charge exerts 
 ^1 upon another according to the prevaihng 
 form of the electromagnetic theory. Let us at first 
 suppose that the motion of one of the two charges being 
 considered is circular motion, and that the other charge 
 is at rest. We may form a definite picture by imagin- 
 ing that the stationary charge is positive and represents 
 the nucleus of some atom fixed in a photographic plate 
 ready to receive the radiation from a small amount of 
 hydrogen gas confined in a vacuum tube some distance 
 away. The charge moving in the circle may then repre- 
 sent a single negative electron in one of the atoms of the 
 hydrogen gas. Whether or not the photographic plate 
 shows anything upon development will depend upon the 
 mechanical force that has acted upon this one atomic 
 nucleus, which really represents them all and thus repre- 
 sents the whole photographic plate. 
 
 The equation expressing this instantaneous force acting 
 upon one revolving electron due to a second electron has 
 been developed in full, and the so-called electric com- 
 ponent of it published in equations (48), (49), and (50), 
 pages 453, 454, of the Physical Review for June, 191 7. 
 Fortunately this equation has been checked by Dr. G. A. 
 Schott in an article in the Physical Review for July, 191 8, 
 where, on page 23, he remarks, " The following investiga- 
 tion is based on Crehore's equations for the electric part 
 of the mechanical force {loc. cit., pp. 453, 454), which 
 
 30 
 
The Atom 31 
 
 have been verified, except some obvious misprints, 
 e.g., 02 for Ci in the last term of (49)." By making the 
 radius of the orbit of the first electron, Oi, equal to zero, 
 so that the electron is brought to rest, and by changing 
 the sign of the equation so as to make the stationary 
 charge positive instead of negative, this equation is 
 strictly applicable to the case we have chosen as an ex- 
 ample above. The distance between the positive nucleus 
 at in the photographic plate and the center of the 
 orbit of the revolving electron 0' in the hydrogen gas 
 is supposed to be fixed or constant and is represented by 
 r. An inspection of the equation referred to shows that 
 there are some terms in it which vary as the inverse 
 first power of r, and others as the inverse square and 
 higher powers. 
 
 Now, any term which varies as the inverse first power 
 of the distance becomes immensely greater than terms 
 which vary as the inverse square or higher powers of the 
 distance if the distance r is taken large enough. Let 
 us pick out, therefore, from this equation only those 
 terms which vary as the inverse distance and write them 
 down separately. It must be stated, however, that we 
 are at liberty to choose the i, j, and k axes in any direc- 
 tions we please because the charge ei is now supposed 
 to be stationary, and it can, therefore, make no difference 
 in the force upon it how this electron is oriented. In- 
 deed it lost its power of orientation as soon as it ceased 
 to revolve in an orbit. Let us, then, take the fe-axis 
 along the line joining the centers, namely the line 00'. 
 The angle a is then the angle between the direction of 
 this fe-axis and the axis of revolution of the negative 
 electron. The coordinates, x, y, and z, of the equation 
 locate the position of the point 0' with reference to 0, 
 along the i, j, and k axes respectively. Since we have 
 
32 The Atom 
 
 now located the point 0' on the fe-axis, both x and y 
 are zero. Hence, putting x = y = ai = o in this equa- 
 tion, and retaining only terms varying as the inverse 
 first power of r, we obtain as the electric force 
 
 ciea ^i' 
 
 eiEk = o (8) 
 
 The letters S2 and C2 are abbreviations for the following : 
 
 eiEi = ^^'^ {cos a) Sii, (6) 
 
 6162 ft"^ . , , 
 
 52 = sin CO2 U - - j + 02 (9) 
 
 C2 = cos coaf f ) + 62 , .... (lo) 
 
 these quantities being functions of the time. The letter 
 A stands for the quantity 
 
 ^ = 1-7^. (11) 
 
 and this may be considered to be equal to unity for our 
 present purpose, since q^ is small compared with c; that 
 is, the velocity of the electron ca is small compared with 
 the velocity of light. 
 
 In these equations it is found that there are no terms 
 in the fe-component of the force that vary as the inverse 
 distance, and hence the force in (8) is put equal to zero, 
 the meaning being that the force is very small by com- 
 parison with the forces along the i and j axes, and that 
 this small force varies as the inverse square of the dis- 
 tance instead of the inverse first power. When the 
 distance 00' is large compared with the radius of the 
 orbit of the revolving electron at 0', then the instanta- 
 
The Atom 33 
 
 neous distance R between the point and the instanta- 
 neous position of the electron becomes very approximately 
 equal to r, the distance between centers which is fixed, 
 and we may write very approximately the sum of the 
 three forces (6), (7) and (8) as follows: 
 
 ei£ = — ^'C(cosa)52{ + C2i] (12) 
 
 T G2 
 
 Let us examine this equation and put into words some 
 of the statements that it implies. The quantities out- 
 side of the bracket are constants. These are the fixed 
 distance between and 0', namely r; the fixed electrical 
 charge on the electron ea which is revolving, and the 
 fixed electrical charge on the stationary nucleus d; the 
 constant speed of the moving electron /Sa, expressed in 
 terms of the velocity of light as a unit, and the fixed 
 radius of the orbit of 62, namely 02. Within the bracket 
 the 52 and C2 are simple harmonic functions of the time, 
 according to (9) and (10) above, regarding R as equivalent 
 to r. There remains only the angle a, which denotes 
 the angle between the line 00' and the axis of revolution 
 of the electron 62. When this angle is zero, the axis of 
 revolution coincides with the line 00', and the plane of 
 the orbit is then perpendicular to the line joining centers, 
 and the orbit, when viewed from 0, appears as a circle. 
 In this position cos cc = i, and the force expressed by 
 (12) is then a purely circular force made up of two har- 
 monic components at right angles to each other. This 
 whole force lies in the i-j plane perpendicular to the 
 line 00', and is also in this instance parallel to the plane 
 of the orbit of 62. 
 
 If the plane of the orbit of 62 were turned through a 
 right angle so as to contain the line 00', the orbit when 
 viewed from would appear as a straight line instead 
 
34 The Atom 
 
 of a circle. The angle a would be a right angle, and its 
 cosine be equal to zero, so that the i-term vanishes from 
 equation (12). The force then becomes a simple har- 
 monic force, still being in the plaiie perpendicular to 00', 
 however, but in one straight line only, parallel to the axis 
 of j. For any position of the orbit of 62 intermediate 
 between these two extreme positions just supposed, the 
 cosine of a is less than unity and greater than zero, and 
 the orbit, as viewed from the point 0, appears as an ellipse. 
 The force upon ei then has two harmonic components at 
 right angles to each other of the same period but dif- 
 ferent amphtudes, the amphtude along the i-axis being 
 less than that along the j'-axis. The force then becomes 
 an eUiptical force still acting in the plane perpendicular to 
 the Kne 00'. 
 
 Now the equation expressing circular motion of the 
 electron 62, from which this force equation has been de- 
 rived, is as follows: 
 
 ra = a2[(cos a)Sii + dj + (sin a)Sik'] (13) 
 
 By a first differentiation of this with respect to the 
 time, the vector velocity of the electron is obtained as 
 follows: 
 
 92 = a2aj2[(cos a)Cii - 52^ + (sin a)Cik'] (14) 
 
 And by a second differentiation, the vector accelera- 
 tion of the electron £2 is as follows: 
 
 /2 = - oaWaKcos a)Sii + Qj + (sin a)S2fe]. . . . (15) 
 
 The coefficient of (15) may be written in terms of 182 for 
 comparison with (12) above because of the relation of 
 definition 
 
 o osWz , ,182^ , ^\ 
 
 P2 = , or a2W2'=c2— • (16) 
 
 c O2 
 
The Atom 35 
 
 Whence, we have as an equivalent of (15) 
 
 /2 = - c^— [(cos 0)821 + Cij + (sin a)Si}C\. . . . (17) 
 
 It is at once apparent from a comparison between this 
 acceleration and the expression for the mechanical force 
 upon the stationary nucleus, (12), that the portion within 
 the bracket is exactly the same in the two expressions, 
 except that the fe-component is missing in the force equa- 
 tion. Except for a certain constant multipher the ex- 
 pression for the force is evidently equal to the portion of 
 the acceleration which is resolved in the i-j plane, per- 
 pendicular to the line 00'. 
 
 Let us, therefore, denote by /;,- the sum of the i and 
 the j components of the acceleration only, and write 
 
 h = - c^ -[(cos a)52i + C23]. . . . (18) 
 
 If this is multiphed by the quantity eie^/ch, we obtain 
 
 !fV«=--^[(cosa)52i + Gj-]. . . . (19) 
 c'r r 02 
 
 But this is exactly equal and of opposite sign to the ex- 
 pression for the mechanical force in (12). Hence we may 
 equate the first members and obtain the equation 
 
 e.E^-'p, (20) 
 
 In words this equation states that the mechanical 
 force eiE, acting upon the stationary nucleus d of the 
 atom in the photographic plate at 0, due to the revolving 
 electron about the point 0' in the hydrogen gas, is pro- 
 portional to the acceleration of the revolving electron 
 when resolved in the plane perpendicular to the line 00', 
 but that it has the opposite sign or direction. It does 
 
36 The Atom 
 
 not matter what the acceleration of the moving electron 
 may be in the direction of the line of centers 00' so far 
 as the mechanical force is concerned, for it is of no effect. 
 Thus far we have dealt with but one component of 
 the mechanical force, namely the electric component, 
 eiE. To this must be added the magnetic component in 
 order that the result shall be perfectly general. The 
 complete expression for the mechanical force in the 
 current form of electromagnetic theory due to Larmor 
 and Lorentz is as follows: 
 
 F = ei\E + ^ giXHj (21) 
 
 where qi is the velocity of the first charge upon which we 
 are getting the force. In the present example this charge 
 is the stationary nucleus of the atom in the photographic 
 plate, which is supposed to have no velocity, so that 
 qi = o, and the magnetic component of the force is, 
 therefore, zero. The result given above in (20) is, there- 
 fore, perfectly general, and requires no modification, the 
 whole mechanical force being due to the electric com- 
 ponent of the force. 
 
 Let us next discuss this equation (20) more fully. It 
 follows directly from this result that the statement is 
 true whether the motion of the electron in the hydrogen 
 gas is circular or not, namely, that the force upon the 
 fixed nucleus is proportional to the acceleration of the 
 moving electron resolved in the plane perpendicular to 
 the line 00'; for, had we assumed at the beginning that 
 the motion of the moving electron was compounded of 
 two simple circular motions having independent periods 
 and amplitudes, we should have arrived at the conclu- 
 sion that the force upon the stationary nucleus is merely 
 proportional to the sum of the two independent accelera- 
 
The Atom 37 
 
 tions of the electron resolved in the i-j plane. From 
 this we may immediately infer that the theorem is gen- 
 eral for any kind of motion whatever of the moving 
 electron, for this complex motion may, by means of the 
 well-known theorem of Fourier, be resolved into a series 
 of simple harmonic motions and a series of simple har- 
 monic accelerations. 
 
 If this is true, it ought to be possible to give a more 
 general proof of it by going back to the fundamental 
 equations of electromagnetic theory without making any 
 assumption as to circular motion. And, indeed, this 
 proof of the theorem has been obtained in this manner 
 directly from the more fundamental equations, but it is 
 not given here partly because of the space required, as 
 well as the necessity for the introduction of the vector 
 notation in which these fundamental equations are ex- 
 pressed. The proof above given is, however, just as 
 general and is easily grasped. It rests, however, upon 
 the assumption that the equation for the force with cir- 
 cular motion of the electrons has been obtained without 
 error from the more general equations. Since this equa- 
 tion has been checked by G. A. Schott, as stated, it is 
 safe to assume that it is correct on the premises. 
 
IV 
 
 ' E are now in a position to examine into some 
 of the consequences of this theorem, which 
 may be regarded as established, as applied to 
 the case before us, namely the hydrogen gas 
 radiating its energy so that a portion of it is received by 
 the atoms in the photographic plate. We shall postulate 
 at the beginning that what we see when the plate is 
 developed after exposure is in a sense merely a record of 
 the energy that atoms in the plate received during ex- 
 posure to the radiation, and need not concern ourselves 
 at present with the obscure processes by which this 
 energy is revealed to us through the process of develop- 
 ment of the plate. We shall also consider that this 
 energy is some function of the mechanical force acting 
 upon the nuclei of the atoms of the plate during exposure. 
 Each atom in the plate is, of course, acted upon by a 
 large number of the electrons in the distant hydrogen gas, 
 and the force that any one atom in the plate experiences 
 may be regarded as proportional to the resolved sum of 
 the accelerations of all the electrons in the hydrogen gas 
 which are brought to bear upon it. 
 
 Let us, first, therefore, give some consideration to the 
 normal state of this gas, in which it is supposed that all 
 the orbits of the electrons are true circular paths. If 
 these orbits all have the same radius and if the revolution 
 of the electrons is at the same speed in all, then the sum 
 of the forces due to them all acting upon the atom at 0, 
 being a vector sum and approximately in one plane per- 
 
 38 
 
The Atom 39 
 
 pendicular to the line joining the photographic plate 
 and the hydrogen gas, would probably be very small, 
 because the planes of the orbits of the electrons in the 
 atoms of hydrogen are turned in every possible orienta- 
 tion, and there would be a tendency to cancellation of 
 the force. To obtain a rigid proof that the force would 
 be exactly zero under these circumstances it would be nec- 
 essary to allow for the slight differences in the distances 
 of the centers of the orbits in the hydrogen from the 
 nucleus of the atom at 0. The problem under these 
 assumed conditions becomes a statistical question that 
 would require treatment by the theory of probabilities, 
 and presents some difficulty. 
 
 The difficulty woufd be much greater if we should 
 imagine that an infinite number of possibfe stable orbits 
 exists, each efectron having a frequency of revolution 
 corresponding to the particufar orbit, in the normal 
 neutral state of the hydrogen. The chance that all of 
 the force would cancel woufd be far smaller. They 
 shoufd, however, exactfy cancel in order to agree with 
 observation; for the normal gas emits no characteristic 
 radiation and does not affect a photographic pfate. There 
 is a difficufty here in supposing that a farge number of 
 different-sized orbits can exist in the normal state of the 
 gas. And, if they do not exist in the normaf state, there 
 is no utility in supposing that they ever exist at all. 
 
 All of these difficufties disappear when it is assumed 
 that each hydrogen atom has two electrons instead of 
 one in its normal state, these electrons being located at 
 the opposite ends of a common diameter of the orbit; 
 for, then, the sum of the acceferations of the two efec- 
 trons in each atom is exactfy zero, and of course the total 
 force upon the nucleus of the atom at is afways zero 
 as long as the etectrons in the hydrogen foffow a purefy 
 
40 The Atom 
 
 circular path. There is then no necessity to attempt a 
 proof of the statistical theorem just proposed for the 
 single electron atom. This fact in itself supplies an ad- 
 ditional reason to those obtained from other considera- 
 tions for supposing that the hydrogen atom has two 
 instead of a single electron. We shall throughout this 
 work assume that the hydrogen atom has two electrons 
 in its normal state. 
 
 It has now been shown that hydrogen gas in its normal 
 undisturbed state will produce no force upon any atom 
 in the photographic plate, and will behave as though no 
 energy is being radiated, a circumstance that is in com- 
 plete agreement with observations. If we now begin to 
 disturb the normal condition of the gas by bombardment 
 with electrons from an external source, or by alpha 
 particles, then the effect may be pictured by supposing 
 that some of the electrons in some of the atoms are driven 
 away farther from the nucleus of their atoms than is 
 normal, following paths that are more complicated than 
 the purely circular orbit, passing out to a maximum dis- 
 tance, which depends upon the amount of energy ab- 
 sorbed, and returning again to the same normal circular 
 motion after a very brief time. Some of the electrons 
 may even be driven completely away from the nucleus, 
 thus ionizing the gas. It is not supposed that every 
 atom in the gas is affected simultaneously and con- 
 tinuously, but that first one and then another is hit, so 
 to speak, its electrons being driven out and returning 
 immediately before the next hit is registered on this 
 particular atom. The apparently continuous bombard- 
 ment is really distributed among a large number of atoms 
 of the gas, no one of them experiencing a continuous im- 
 pulsive force. 
 
 It is our purpose to inquire into the nature of the paths 
 
The Atom 4i 
 
 described by the electrons in the hydrogen atoms in re- 
 turning to their normal state after being disturbed. 
 The chief guide must be the known spectrum of hydro- 
 gen and an application of the theorem obtained above 
 from electromagnetic theory to the case. Any admissible 
 form of path must be such that only those accelerations 
 are admissible which have periods equal to those in the 
 observed spectrum. In this mode of looking at the 
 subject we are reversing the common procedure, which 
 assumes as a starting point the law of force acting and 
 attempts to work out from it the paths of the electrons. 
 We do not assume any law of force, but leave that to 
 come out as a final product instead of an initial as- 
 sumption. *• 
 
 Let us first place before us the whole spectrum of 
 hydrogen for constant reference. This element is selected 
 for one reason because the atom of hydrogen is probably 
 the simplest in structure of any of the atoms, and for 
 another because the complete spectrum of hydrogen is 
 known with great probability all the way from the lowest 
 possible frequency up to the highest, as we shall see, 
 although the total number of lines in the spectrum is 
 infinite. Fortunately in the case of hydrogen a mathe- 
 matical formula has been found that makes it possible 
 to express in extremely simple language all of the infinity 
 of lines in this spectrum. This cannot yet be said to 
 be true of many of the other elements. The spectrum of 
 hydrogen is ,shown in Fig. i. The lines shown in this 
 chart have been observed experimentally. The visible 
 portion of the spectrum lies between the points indicated 
 approximately, and only a very limited number of lines 
 in the Balmer series lie in this region. If we knew only 
 the spectrum in the visible region, it is evident that no 
 important generalization could have been obtained from 
 
42 The Atom 
 
 it. All of the lines that have been observed may be 
 very accurately expressed by the formula 
 
 In this formula v' denotes the frequency of the vibra- 
 tion corresponding to the line which is indicated by its 
 wave-length in the figure, i^ is a constant equal to 
 3.290 X 10'* and known as Rydberg's constant, which is 
 
 • HYDROGEN SPECTRUA 
 
 s 
 
 K Visible 
 
 I ■ I 1-! 1 ifi — 1 1 1 — 
 
 
 ■H 1 f— f 
 
 «i32r, a'765 4. r, 3 «> '09 8 7 
 
 V 
 
 ' - • / »^ 
 
 LYMAN BAintl^ >f 
 
 n-i 7-2=2 
 
 Fig. I. 
 
 Vj 
 
 one of the most important constants we know of in 
 nature. It appears not only in the spectrum of hydro- 
 gen but in the spectrum of all of the other elements 
 whose spectra have been expressed by a formula, t^ and 
 Ti are simply integers, and may have any values whatever, 
 provided only they are not such as to make the fre- 
 quency have a negative value. 
 
 Let us first look at this formula in the usual way and 
 afterwards consider the modification of it that we shall 
 require in the interpretation of the emission of this spec- 
 trum by the atoms of the gas. If we set T2 = i, and then 
 give to Ti values 2, 3, 4, etc., to infinity, we obtain the 
 
The Atom 43 
 
 frequencies of all the lines of the so-called Lyman series, 
 which has the greatest frequencies and shortest wave- 
 lengths and is seen at the extreme left in Fig. i cramped 
 up between comparatively narrow limits. If we set 
 T2 = 2, and give to Ti the successive values 3, 4, 5, etc., 
 to infinity, we obtain the series known as the Balmer 
 series, part of which appears in the visible spectrum in 
 the central portion of the figure, the rest extending into 
 the ultra-violet. If we set T2 = 3, and give to Ti the 
 
 WAVE - LENGTHS 
 
 rj rO ;^ «^ MS r-~-, oQ 0% 
 
 _+ 4: + .^ 5: :^ 4= ~ — 
 
 y 
 
 Y 
 
 PASCHErtt 
 
 Fig. I. 
 
 values 4, 5, 6, etc., to infinity, we obtain the series 
 known as the Paschen series at the right in the figure. 
 All of this series is in the infra-red portion of the 
 spectrum. 
 
 A sufficient number of lines has thus been observed to 
 make it quite certain that the formula (22) is general, 
 that is to say, we may predict from it that, if we should 
 set T2 = 4, we should obtain another similar series of 
 lines which have not yet been observed experimentally 
 further off in the infra-red beyond the Paschen series, 
 and so on for values of T2 = 5, 6, 7, etc., to infinity. 
 The frequencies grow smaller and smaller and the wave- 
 
44 The Atom 
 
 lengths greater and greater as we advance in this 
 direction. 
 
 The X-ray spectrum of hydrogen has never been ob- 
 tained directly by the use of these rays. There are grave 
 difficulties in getting the X-ray spectra of the elements 
 of low atomic number. There are reasons to believe 
 that the X-ray spectrum of hydrogen will never extend 
 the spectrum to shorter wave-lengths than those given 
 in the Lyman series, and that in this series we really have 
 the X-ray spectrum of hydrogen. 
 
 The whole spectrum that has just been described is 
 produced by the sum of the effects of all of the atoms in 
 the gas. How much of it is contributed by a single atom 
 at any one time, or during any one excursion of its mo- 
 tion, is not, of course, revealed by any knowledge of 
 the summation of the effects of them all. The Bohr 
 theory above mentioned attributes the production of but 
 a single line in the spectrum to one operation of one elec- 
 tron in changing over from one stable orbit to another. 
 In the way that we prefer to view this question a whole 
 series of spectral hnes is emitted by the two electrons in 
 the atom in one operation while they are returning again 
 to their original orbit after being displaced. But this 
 series of hnes is neither the Lyman, Balmer, Paschen, 
 nor any of the series commonly spoken of. Assuming, 
 for the sake of forming a definite picture, that the two 
 electrons return to the original orbit in a species of spiral 
 paths approaching rapidly at first and then slower and 
 slower as they get closer and closer to the final orbit, it 
 appears that the sum of the accelerations of these two 
 electrons should contain frequencies which change rapidly 
 at first and then slower and slower, approaching zero as 
 they finally attain their steady orbit. The frequencies 
 in, say, the Balmer series, however, are crowded close 
 
The Atom 45 
 
 together at a head and separated more and more as we 
 go down in the series to lower frequencies. According 
 to this way of viewing the matter, these series are not so 
 well adapted to the case as if we looked at equation (22) 
 in a different way. 
 
 The following equation is exactly equivalent to (22) 
 and has the advantage that the spectrum is divided up 
 into series in a manner more suited for our purposes. 
 
 >' = K(i-,-^.) (.3) 
 
 If in this we set T2 = 1, and give to t the values i, 2, 3, 
 etc., to infinity we obtain the first hnes of lowest fre- 
 quency of every one of the former series, namely the 
 Lyman, Balmer, Paschen, etc., series. If we set T2 = 2, 
 and give to t the values i, 2, 3, etc., to infinity, as be- 
 fore, we obtain the second lines of each of the former 
 series. And so on, by setting T2 = 3, 4, 5, etc., to in- 
 finity, we obtain eventually all of the lines that were 
 obtained before. When T2 = 0°, all of the so-called 
 " heads " of the Lyman, Balmer, Paschen, etc., series are 
 obtained. 
 
 An inspection of the formula shows that, as t becomes 
 larger and larger, the two terms within the parenthesis 
 approach each other in value continually, and their dif- 
 ference, therefore, approaches zero in every case, but 
 very slowly. That is to say, the lines in these series are 
 crowded nearer and nearer together at the low-frequency 
 end of the series. The wave-lengths, however, do not 
 appear so crowded since they are proportional to the 
 reciprocals of the frequencies, and an extremely small 
 difference in a small frequency may make a large ab- 
 solute difference in the' wave-length. 
 
 As stated above, we are about to discuss possible forms 
 
46 The Atom 
 
 of the paths followed by the electrons in returning to 
 their original orbit after being displaced. It will, of 
 course, be understood that there is not sufficient data 
 supphed by a knowledge of the spectrum alone to make 
 this problem susceptible of but one solution. We have 
 supposed that the spectrum supplies us with a knowledge 
 of the frequencies only that enter into the sum of the 
 accelerations of the electrons in their motions in the gas 
 while radiating energy. Further experimental data as to 
 the amplitudes of these various harmonic components of 
 the accelerations is necessary. A knowledge of a sum is 
 not as satisfactory as a knowledge of each of its com- 
 ponent parts, the accelerations of the individual elec- 
 trons, because a sum may be made up in an infinite 
 variety of ways. But it is the best knowledge we have 
 at present, and the only way that is open is to make 
 the most reasonable assumption that we can concerning 
 the individual parts that is always in complete accord 
 with the knowledge we possess of the sum, and see whether 
 the results derivable from the assumptions are in com- 
 plete harmony with the known facts of observation. 
 It is most desirable to have experimental checks at as 
 many points as possible in order that the assumptions 
 made may be tested in a variety of ways. One experi- 
 mental check is afforded by the agreement between the 
 observed values of the voltages required to ionize the 
 hydrogen gas, that is, to separate some of the electrons 
 completely from their atomic nuclei. It is shown in a 
 later section that the theory here presented gives all of 
 the ionizing voltages to within the experimental error 
 that have been observed in hydrogen gas. The Bohr 
 theory gives no indication of an ionizing voltage at 15.8 
 volts . recently observed. The lowest voltage that the 
 Bohr theory gives is 10.15 volts, but ionization does not 
 
The Atom 47 
 
 begin until the voltage is above 1 1, according to the most 
 reliable experimental results. This theory gives the 
 minimum value as 11. 132 volts and the maximum value 
 as 15.5 volts, which is in better agreement with the 
 observations. 
 
apology is, therefore, offered for writing on 
 this subject in a suggestive and general man- 
 ner. The fundamentally new point in the 
 theory is that a whole series of spectrum 
 ines are produced in one operation of the electrons in re- 
 turning by a species of spiral paths to their original stable 
 orbit, and that there is but one orbit of a uniform size in 
 the normal hydrogen atom. It is not intended to imply 
 by this assertion that all atoms of one element are always 
 exactly alike in their neutral condition. Where there are 
 a multiplicity of rings of electrons, as in most of the more 
 complex atoms, it is very likely that there are a number 
 of different possible stable configurations dependent upon 
 the order of the rings as we proceed outward from the 
 nucleus. If, for example, a ring of three electrons should 
 interchange places with a ring of four, it is held that this 
 would not affect many of the properties of the atom, such 
 as its weight and atomic number, but that it may affect 
 other properties such as its chemical valency. So long 
 as the weight and atomic number remain unchanged, 
 such an atom is called by the same name, but many 
 atoms having the same name are known to behave dif- 
 ferently in combining with other atoms on different 
 occasions. In the case of hydrogen, however, no in- 
 terchange of rings is possible because there is but one. 
 
 It is of course possible that the precise paths followed 
 by the electrons in hydrogen, which we are about to 
 describe, may not be the actual paths, since the problem 
 admittedly has a number of possible solutions on the 
 
 48 
 
The Atom 49 
 
 limited amount of experimental data now available, but 
 it seems much more desirable to make definite assump- 
 tions, which lead to definite results, even if future 
 experimental data shall compel a revision of these as- 
 sumptions, than it is not to make such assumptions. 
 For, with definite assumptions, it is possible to draw defi- 
 nite conclusions and then to compare these with the facts 
 of observation so far as possible. Unless some contradic- 
 tion between the assumptions and experimental data is 
 found we will then have at least one possible solution, 
 which is worth something. 
 
 Let us, therefore, denote by v, 2ir times the frequency 
 of a particular line in the spectrum, v is then an angular 
 velocity as follows: 
 
 ' = '''^{i-FTl^^ ^'^^ 
 
 It is assumed that in any one motion of the electrons 
 in the hydrogen atom T2 is a fixed integer dependent upon 
 the amount of energy that the system has absorbed from 
 an external source, but that t takes all possible values 
 at once from i to infinity. Let us denote by p2 and pi 
 the position vectors of the two electrons e^ and ei as they 
 move about, and take the hydrogen nucleus as the origin 
 of these vectors. Then the vector velocities of these elec- 
 trons are denoted by -r^ and -^p, and the vector accelera- 
 tions by -^ and -J^ ■ Let us now assume that the sum of 
 
 the two vector accelerations of the two electrons in one 
 atom during one excursion is represented by the equation 
 
 ^ + ^= fer.T { vh-"' [(sin vt)i + (cos vt)n ] . . (25) 
 
 in which kr, remains constant during the whole motion, 
 
50 The Atom 
 
 t represents time and v is the expression in (24). The 
 summation means that we are to write a term hke 
 that in the brace for every value of v corresponding to 
 values of t from i to infinity, so that the equation is an 
 infinite series of terms. Both kr^ and v are functions of T2, 
 which remains fixed during one excursion, but differs 
 for different excursions on different occasions. Hence 
 the equation is different for each different amount of 
 energy absorbed by the system. According to the 
 theorem above established, the force that these two elec- 
 trons in the one hydrogen atom exerts upon the nucleus 
 of the atom at in the photographic plate is proportional 
 to the sum of the accelerations of the electrons resolved 
 in the i-j plane, that is, a plane perpendicular to the line 
 joining the nuclei of the two atoms. With a different 
 constant multiplier this equation (25), therefore, repre- 
 sents the force that the one hydrogen atom contributes 
 during one single excursion to the formation of the spec- 
 trum of hydrogen. This is entirely consistent with the 
 observed spectrum because the only frequencies con- 
 tained in the force equation are those which are observed 
 in the resulting spectrum of hydrogen. These fre- 
 quencies are, however, infinite in number and consist of 
 all the first lines, say, of the Lyman, Balmer, Paschen, 
 fourth, etc., series, or of all of the second, third, etc., 
 lines of these series on different occasions. A multitude 
 of hydrogen atoms, each receiving a different amount of 
 energy, will accordingly produce the whole spectrum of 
 hydrogen, and the same atom at a later time may give a 
 different series of lines. 
 
 It will be noticed that there is no fe-component in the 
 acceleration (25). This is merely because the spectrum 
 tells us nothing about it, and the equation merely repre- 
 sents the component of the acceleration resolved in the 
 
The Atom 51 
 
 i-j plane. The fe-component may be assumed to be any- 
 thing whatever without having any effect upon the 
 spectrum produced. Let us imagine, therefore, that the 
 whole motion of the electrons takes place in the one plane 
 perpendicular to the line 00'. 
 
 Each term of this infinite series in (25) represents a 
 purely circular motion, the part within the bracket, 
 affected by an exponential factor e""', which causes the 
 amphtude of the motion to diminish with time finally to 
 zero. Hence each term separately becomes zero after 
 an infinite time, and so the whole acceleration vanishes, 
 as it should. For then it is considered that the two 
 electrons are located at the opposite ends of a common 
 diameter in a fixed circular orbit, whence the sum of the 
 accelerations of the two is evidently zero. 
 
 Upon integration of this equation with respect to the 
 time the sum of the velocities of the two electrons is 
 obtained, namely: 
 
 dp dp^ KY f j;2e-.«[(_ sin vt - cos vt)i 
 
 dt dt 2 T=i I '-^ 
 
 + (- cos vt + sin vt)j] \ . . . . (26) 
 
 No constant of integration is added because, after an 
 infinite time, when the two electrons are in their final 
 orbit, the sum of their velocities must evidently vanish. 
 
 A second integration gives us the sum of the two 
 position vectors of the two electrons, namely: 
 
 P2 + Pi = -j^J" I ye-Kcos vt)i - (sin vt)j} } . (27) 
 
 No constant of integration is added here either, be- 
 cause, after an infinite time, when the two electrons are 
 in their final orbit at opposite ends of a common diameter, 
 evidently the sum of their position vectors must be zero. 
 
52 The Atom 
 
 It will be noticed that the factor f ', which was introduced 
 into the equation for the acceleration (25), is reduced 
 by the first integration to p' in (26) and is reduced again 
 by the second integration to v in (27). The question 
 why we need any factor v in the equation (27) requires 
 some explanation. That is, why could we not have had 
 I*" in (27), v^ in (26), and y^.in (25)? Some such mul- 
 tiplying factor is required in (27) as we may see by making 
 the time equal to zero. Without any factor of this kind 
 each term of the infinite series becomes equal to unity, 
 for the sine of zero vanishes, the cosine becomes unity, 
 and the exponential becomes unity. The sum of the 
 series therefore becomes the sum of an infinite number 
 of units and is infinite and not finite. The presence of 
 the factor v, however, makes the series finite, as we shall 
 see. There is, of course, no proof that this factor should 
 be simply v; but we cannot seek for proofs throughout 
 this investigation until some particular example of it is 
 completed so that comparisons may be made between the 
 results of the assumptions and the known experimental 
 facts. 
 
 Although this equation, giving the sum of the two 
 position vectors, is of some interest, and we shall discuss 
 it more fully later, yet we much prefer to know how this 
 sum may be divided up into its two components, p2 and 
 Pi. So long as the sum is not changed, we are evidently 
 at liberty to divide it up in a number of possible ways, 
 but probably only one of these ways is admissible. 
 
 There is reason to think that the quantity 
 
 plays some part in the individual motions of the electrons 
 as well, as in the difference of their position vectors 
 
The Atom 53 
 
 Pi — Pi- This quantity /j. is precisely the same as v, 
 except that the sign of the second term in (24) is positive 
 instead of negative. It is also thought that the sum of fj. 
 and V is involved in the individual motions of the elec- 
 trons. We shall make the arbitrary assumption that the 
 vector difference of the accelerations of the two electrons 
 has the following expression: 
 
 d^P2 
 
 
 
 T = « 
 
 -Br, -2 
 
 ^3g-M([(sin fJLt)i + (cos fit) J} 
 (M + j')'e-f''+'')l(sin {/ji. + v)t)i 
 
 + (cos (m + p)t)f] \ - A(4xK')''[(cos 4TrKt)i 
 
 - (sin 4irKt)f] (29) 
 
 The first summation in this is exactly analogous with 
 equation (25), giving the sum of the accelerations, ex- 
 cept that fi replaces v. The second summation is an 
 entirely analogous summation in which fi + v replaces 
 the V of equation (25), and a new constant, Bt^, replaces 
 the ferj. The last term is not a summation but a single 
 term with a new constant A. This term may be re- 
 garded, if we please, merely as the first term of the pre- 
 ceding summation in which the exponent of the Naperian 
 base e is zero, and so it does not appear. This last 
 term is the only one which is left when the time is infinite, 
 because both of the summations vanish, due to the ex- 
 ponential factors. The difference of the accelerations in 
 the final orbit is of course not zero Hke their sum, but is 
 represented by a purely circular motion, namely the last 
 term in the equation. 
 
 Assuming this equation to represent the difference of 
 the vector accelerations, we may find the difference of 
 the velocities by a first integration as follows : 
 
54 The Atom 
 
 + (- COS jUt + sin ixt)f} > 
 
 - — "^S" f (m + y)2e-(*'+'')l(- sin (/x + v)t 
 
 - COS ifJL + v)t)i+ (- COS ifJL + v)t + sin (/x + j')Oj] \ 
 
 - A{^TrK)[(^sm 4,TrKt)i + {cos ^TrKt)]} (30) 
 
 No constant of integration is added in this case either, 
 because of the known final condition. The difference of 
 the velocities must be double the velocity of each electron 
 and be represented by a circular motion such as is given 
 by the last term of this equation. 
 
 A second integration gives the difference of the posi- 
 tion vectors as follows: 
 
 Pa - pi = — ' S < /ie~'"[(cos iit)i - (sin nt)if] > 
 
 - -j^l" { (m + J')e-(^+''>'[(cos (m + v)t)i - (sin (/x + v)t)j"\ \ 
 + A[_{cos 4.irKt)i - {sm 4.irKt)i\ (31) 
 
 No constant of integration is required here either, 
 because of the known final condition of the motion. 
 The difference of the position vectors must be equal to 
 the diameter of the orbit and be represented by a purely 
 circular motion as is given by the last term of the equation. 
 
 The accelerations, velocities, and position vectors of 
 the two electrons individually may now be obtained by 
 the simple addition and subtraction of the above ex- 
 pressions for the sums and the differences giving the 
 following equations. 
 
The Atom §§ 
 
 J = j'l" { vh-'"\:{sm vt)i + (cos vt)f] } 
 ± ^^2°° I M^e-^'CCsin iJ,t)i + (cos mOjI } 
 =^ — ''^ir ( (m + J^)%-f''+'''l(sin (m + J')Oi 
 
 2 T=I I 
 
 + (cos(M+J')Oi]| 
 
 =F= - (47rK')2[(cos47rifOi - (sin47rK'0i] (32) 
 
 4r = —^2°° ( J'V'"[( - sin i/J - cos vt)i 
 at 4 T=i I. 
 
 + (- cos vt + sin vt)j] > 
 
 kr ■^=°° f 
 ± — ^ 2 < )LiV"[(- sin juf - cos iJLt)i 
 4 T=i L 
 
 + (- cos Mf + sin ;uOi] r 
 
 =F — "'^ir ( (m + ?')%-(''+'')'[(- sin (m + !')« 
 
 4 T = I I 
 
 - cos(/x + v)t)i + (- cos (/x + J')f + sin {fj. + v)t)j2 \ 
 =F - {4,TrK)l(sm 4.TrKt)i + (cos 4TrKt)j] (33) 
 
 3 < j'e-'''[(cos I'Oi - (sin vt)j2 \ 
 
 4 
 
 -^^£ ( Me-^Kcos MOi - (sin mOjI 1 
 :^ :?!:^^ir / (f,+p)e-^t^+'')'\:(cos(ti+v)t)i-ismifi+p)t)j} 
 
 4 T = I 1 
 
 A 
 =1= - [(cos 4.TKt)i - (sin 4TKtyj2 (34) 
 
$6 The Atom 
 
 The upper signs should be used for the second and the 
 lower signs for the first electron. The constants kr,, 
 Bt, and A are evidently connected with the initial and 
 the final condition of the motion. In order to determine 
 these constants let us write down the equations from 
 (25) to (34), first assuming that the time is zero for the 
 initial condition and then assuming that the time is 
 infinity for the final condition. We have ^ 
 
 (t+$). = ''^Mi <«' 
 
 (p2 + pi)o=Y2Wi (37) 
 
 - B..S(m + J')']i (38) 
 
 + [-|-'S(M=)+^2(M+'')^-47ri^A]i . . .(39) 
 
 (P2 - Pi)o = [y'2(M) - ^2(/x + v)+ A^i. . . (40) 
 
 (Sl=-^4-iq^^ + [=*=T(2(M')-S(^')) 
 -^'S(M + »')']i (41) 
 
 '■ The 2's occur so frequently that it facilitates the printing to 
 omit the limits t = t and r = 00. Whenever these limits are not 
 expressed in this volume, it will be understood that the limits in- 
 tended are t = i to T = 00. 
 
The Atom 57 
 
 +■ {=.^'[S(M^)-SM]^^'S(M+»')^=-2^i^A}i . (42) 
 
 (p)o = [- ^ (S(m) - SW) - v^S(M + J-) 
 L 4 4 
 
 =^-Ji (43) 
 
 ©+$').=» («) 
 
 (t+'i).= ° <«' 
 
 (P2 + Pl)o= =0 (46) 
 
 - (sin 4irX0J] (47) 
 
 (^' - ^')^ = - (47ri^)A[(sin 4TKt)i 
 
 + (cos 4TrKt)j^ (48) 
 
 (P2 - Pi)=o = A[(cos 4xA^0i - (sin 4'TKt)j] . . (49) 
 
 (J)^=^(4TA:)^^[(cos47rK0i 
 
 - (sin 47rK'0i] (50) 
 
 (J)^=-(47r/O^C(sin4irA:0i 
 
 + (cos 4TrK'0j] (51) 
 
 (P)« = =^ j[(cos47rX0i - (sin 47rK0i]. . (52) 
 
VI 
 
 |S before, the upper signs apply to the motion 
 of the second electron and the lower to that 
 of the first. It appears from these equa- 
 tions that the final motion of the electrons 
 is a simple circular motion at the opposite ends of a com- 
 mon diameter according to (52). Denoting the radius of 
 this orbit by a, the constant A is therefore equal to twice 
 the radius of the orbit, and we have 
 
 A = 2a (53) 
 
 This constant is the same for every value of T2, and the 
 radius is independent of the initial conditions and of the 
 amount of energy absorbed by the system. The fre- 
 quency of revolution in the orbit is equal to 2K, twice 
 the Rydberg constant, namely 2 x 3.290 X lo'^ This 
 is supposed to come about because the A term in the 
 difference-equation (29) is considered to belong to the 
 {fj, + v) series, being that term of it which has the ex- 
 ponential factor e", and in which t = i. If we add /i 
 and V, which are given in (28) and (24), the second term 
 in the parenthesis cancels, and we have for ail values of T2 
 
 li + v = Wl<^ (54) 
 
 When T = I, this gives the angular velocity 4,irK and the 
 frequency 2K. 
 
 Let us next consider the initial conditions when the 
 time is zero. Equation (43) shows that both electrons 
 are located upon the i-axis when the time is zero, there 
 
 58 
 
The Atom 59 
 
 being no j-component in the equation. In order to ob- 
 tain values of the constants kr, and Bt„ let it be supposed 
 that when the time is zero the outermost electron has 
 reached its maximum distance away from the nucleus 
 and is about to return again. This is the moment when 
 the absorption of energy ceases and the radiation of it 
 begins. If this electron is not moving away from the 
 nucleus, the only motion that it can have at this time is 
 a motion in a direction perpendicular to its radius vector. 
 Its radius vector is, as we have just mentioned, along the 
 i-axis at this time. Hence its velocity must be along the 
 j'-axis when the time is zero. The i-component of the 
 velocity in equation (42) may then be equated to zero. 
 We shall regard the second electron as the more distant 
 electron at this time and therefore use the upper signs 
 in (42), which apply to this electron, giving 
 
 - ^^ (S(M^) + i:{v')) + ^s(m + py = 0. . (55) 
 4 4 
 
 Whence is derived a relation between kr, and Br, as 
 follows : 
 
 The whole velocity of ea is then the j-component of 
 (42), which reduces to the simple expression 
 
 The sum of the velocities of 62 and ei is given by (36). 
 Subtracting from this the velocity of d in (57), the 
 velocity of ei is as follows: 
 
 (^')^=-~^S(»'^)Ci+j] + (4^Ka)j. . . . (58) 
 
6o The Atom 
 
 This equation does not admit of the i-component of the 
 velocity of ei being zero, since neither kr^ nor SCi*^) can 
 vanish, and consequently does not admit of the velocity 
 of Ci being perpendicular to its radius vector, which, as 
 we have seen, is also along the i-axis when the time is 
 zero. It is to be supposed that there is no sudden or 
 abrupt change in the direction of motion of the electrons 
 at the time when the absorption of energy ceases and the 
 radiation of it begins, namely at the zero time. The 
 velocity of ci cannot be zero at this time, according to 
 these equations, and it must have some direction. It is 
 also natural to suppose that there is a symmetry between 
 the outgoing and the return motions, and we may ex- 
 pect this symmetry to exist at the zero time. The only 
 two directions that satisfy these conditions are either 
 along the radius vector or perpendicular to it. Since 
 the velocity of Ci cannot be perpendicular to its radius 
 vector, as we have seen, we shall take its direction of 
 motion when the time is zero along the radius vector, 
 namely along the i-axis, and shall equate the j-component 
 of (58) to zero, giving 
 
 and 
 
 ©0= ~ ^'^("'^i = - (4^^«)»- • • • (60) 
 
 Comparing (57) with (60) it appears that the initial 
 velocities of the two electrons have the same value, that 
 of 62 being in the direction of - j and of ei in the direction 
 of - i. Moreover, comparing these values with the co- 
 efficient of (51), which expresses the velocity in the final 
 orbit, it appears that the initial and final values of the 
 velocities are the same. According to this result there 
 
The Atom 6i 
 
 is no change whatever in the kinetic energy of the elec- 
 trons between their initial and final motions. 
 
 By combining the value of the constant kr,, with the 
 ratio of kr^ to Bt^ in (56), the value of Bn is as follows: 
 
 ^-'=2G;mo^l'+sM'/ ^^^ 
 
 Having determined expressions for the three constants, 
 fexj, -Bt, and A, which enter into the equations of motion 
 from (25) on, we might now substitute them in these 
 equations and calculate numerical examples of the motion 
 for some fixed value of T2. The labor connected with 
 this numerical calculation is, however, considerable even 
 for a single example, and it will be deferred until a later 
 section of the work (See Chapter XII). 
 
VII 
 
 ]]HE sums of the various powers of jj. and v 
 and (fj, + v) enter into these equations 
 through the constants kr^ and Bt^, and we 
 shall next give some consideration to the 
 determination of their numerical values, which will be 
 required before further progress can be made in the in- 
 terpretation of the theory above given. Consider first 
 the summation of (/i + v) as given in (54) above, when 
 T takes all values from i to infinity, and denote the sum 
 by si. We have 
 
 The sum of the series of numbers in the parenthesis is 
 known to be exactly equal to ■k^/6. Hence, we have 
 
 r2' 
 
 -T' 
 
 si = /^ttK-z = 4'^K X 1.644,934,066,8. 
 
 (63) 
 
 In a similar manner, denoting by 53 the sum of {fi + vy, 
 and by S3 the sum of {jj, + u)^, we have 
 
 52=S(M+J')^=(47ri«:)2Si=(47riO^(i+^,+^,+I . . .), .(64) 
 
 and 
 
 53=S(M+^)^ = (47r/^)'S^, = (4xK)3(i + l+i +1. • •)■ (65) 
 
 The numerical values of the sums of the series 
 
 *■ i. 1 1 
 
 — H 1 H— + 
 
 3" 4" 
 62 
 
 (66) 
 
The Atom 
 
 63 
 
 have been calculated. A table of these sums to fifteen 
 decimal places and for values of n from i to 35 is given 
 on page 554 of De Morgan's calculus. This table is re- 
 produced here to nine places of decimals and for the even 
 values of n only in (67). By its means we find 
 
 n 
 
 00 J 
 
 2 
 
 4 
 6 
 8 
 
 1.644.934.067 
 1 . 082,323,234 
 1.017,343,062 
 1 . 004,077,356 
 
 10 
 
 1.000,994,575 
 
 12 
 
 1 . 000,246,087 
 1.000,061,248 
 1 .000,015,282 
 
 18 
 
 1 .000,003,817 
 
 20 
 
 1 . 000,000,954 
 
 22 
 
 1 . 000,000,238,45 
 
 24 
 
 1 .000,000,059,61 
 
 26 
 
 1 .000,000,014,90 
 
 28 
 
 1 . 000,000,003,73 
 
 30 
 
 1 . 000,000,000,93 
 
 32 
 
 1 . 000,000,000,23 
 
 34 
 
 1 . 000,000,000,06 
 
 36 
 
 1 .000,000,000,01 
 
 (67) 
 
 (68) 
 (69) 
 
 52 = (4t/0^ X 1.082,323,233,7. 
 
 53 = (47r/0' X 1.017,343,062. 
 Let us next consider the numerical values of the sum 
 
 of V and of fi. We have 
 
 If T2 = 3 say, then 
 Si' = 27r^:(i +i. 
 Ill 
 
 (j + T,yj 
 
 (70) 
 
 I I II 
 
 42 52 6^ 
 
 ■) = 2T/c(l+^, + i). . . (71) 
 
64 
 
 The Atom 
 
 Hence, we may say in general that 
 
 Sj/ = 27rK2 -, 
 
 Similarly, we have the value of 
 
 I 
 
 SM = 27ria;(i+^^_^^^^, 
 
 } 
 
 (72) 
 
 (73) 
 
 and, if Ta = 3 say, we have 
 
 _, „/2 2 2,22, I I I \ 
 
 2M-JirK(-,+j, + -, + j-, + j,+ ...--.-j-l-.j 
 
 = -<f-p-r.-?) <"> 
 
 Hence, generally 
 SM = 27rx(|-Sl) (75) 
 
 It is seen that we obtain (63) by adding (72) and (75). 
 The following table (76a) gives the values of i/t, i/t^ 
 and i/t* for the first ten values of r. And the table 
 (77a) gives the sums of i/t, i/t' and i/t* for the first 
 ten values of t. 
 
 T 
 
 I 
 
 I 
 
 1^ 
 
 I 
 
 7«" 
 
 I 
 
 2 
 
 3 
 
 4 
 5 
 
 6 
 
 9 
 10 
 
 Sum 
 
 1 . 000,000,000 
 0.500,000,000 
 
 0.333.333.333 
 0.250,000,000 
 . 200,000,000 
 
 0.166,6^6,667 
 0.142,857,143 
 0. 125,000,000 
 0. 111,111,111 
 0. 100,000,000 
 
 2.928,968,3 
 
 1 . 000,000,000 
 0.250,000,000 
 0. 111,111,111 
 0.062,500,000 
 . 040,000,000 
 
 0.027,777,778 
 0.020,408,163 
 0.015,625,000 
 0.012,345,679 
 0.010,000,000 
 
 1-549.767.73 
 
 1 . 000,000,000 
 0.062,500,000 
 0.012,345,679 
 0.003,906,250 
 0.001,600,000 
 
 0.000,771,605 
 0.000,416,495 
 0.000,244,141 
 0.000,152,416 
 0.000,100,000 
 
 1.082,036,586 
 
 (76a) 
 
The Atom 
 
 65 
 
 T 
 
 ?;- 
 
 n I 
 f..- 
 
 1 
 
 I 
 2 
 
 3 
 
 4 
 5 
 
 6 
 
 7 
 8 
 
 9 
 10 
 
 1.000,000 
 1 . 500,000 
 
 1-833.333 
 2-083,333 
 2.283,333 
 
 2.445,000 
 2.592,857 
 2.717,857 
 2.828,968 
 2.928,968 
 
 1.000,000 
 1 . 250,000 
 1 .361,111 
 1 .423,611 
 1.463,611 
 
 1.491,388 
 1.511,797 
 1.527,422 
 1.539,768 
 1.549,768 
 
 1 . 000,000,000 
 1 . 062,500,000 
 1.074,845,679 
 1.078,751,929 
 1.080,351,929 
 
 1.081,123,534 
 1 .081,540,029 
 1 .081,784,170 
 1 .081,936,586 
 1 . 082,036,586 
 
 00 
 
 00 
 
 ^2 
 1.644,934 = -g- 
 
 1.082,323,233,7 
 
 ■ (77a) 
 
 Let us next determine the values of ^v^ and ^fi^ in 
 terms of sums of the powers of i /r" so that the table of 
 numerical values of i/t" in (67) may be used for cal- 
 culating Sj*^ and 2)Lt^. We have 
 
 S -7 is given in the table (67) as 1.082,323. If T2 = 3 say, 
 
 T 
 
 then 
 
 ^(TTF* = ? + i-* + i+--- ='-°^"'3^3-S^^ (77) 
 and, generally, 
 
 2(7:p^.= x.o82,323-Si. . . .(78) 
 
66 The Atom 
 
 This leaves the last term of (76) to be evaluated, which 
 may be effected by resolving i/t^(t + t^Y into four 
 partial fractions according to the following identity: 
 
 I 12 I 2 , . 
 
 In summing this expression term by term, T2 is to be re- 
 garded as constant. Since we know that 
 
 2^=? (80) 
 
 and 
 
 S,— i-^, = ^-S-, (81) 
 
 it follows that the sums of the first and the third terms 
 of (79) give 
 
 .7(1-1^) <-' 
 
 Similarly, since 
 
 V, I I I I I I ,„ V 
 
 2f=I + 2 + 3+4+"5+ ^^^^ 
 
 and, when T2 = 3 say, 
 
 S^^=i + -+ (84) 
 
 we find generally that the sum of the second and fourth 
 terms in (79) give 
 
 2 '"» I 
 -TiVr (85) 
 
 By adding (85) and (82) we obtain the complete sum of 
 the last term of (76). The whole sum of (76) is, therefore, 
 
 -Lv^ = (2TrK)^ f 2 X 1.082,323 - ?^ + ±si 
 
 2 ''« I '^' 1 ) 
 
The Atom 
 
 67 
 
 To obtain the sum of fi^ simply change the negative 
 sign in the last term of (76) to positive, giving 
 
 Sm^ = {2TKy \ 2 X 1.082,323 +l~-^,il 
 
 3 ~2 ~2 I T 
 
 — S - - s - . 
 
 Ti' 
 
 / 
 
 . (87) 
 
 By the use of these formulae and the table (67), the 
 following table, (88), of values of Zv^ and S/i^ have been 
 calculated. Let us denote by x, y, and z the following 
 expressions, which are tabulated in (88) and (77a). 
 
 X = 
 
 y = 
 
 2i'2 
 
 
 z =S — 
 
 (89) 
 (90) 
 (91) 
 
 ri 
 
 " [2TKy 
 
 ^ (27rK)2 
 
 
 1 
 2 
 3 
 4 
 5 
 
 6 
 
 7 
 8 
 
 9 
 10 
 
 0.584,910 
 0.832,212 
 
 0.932.793 
 
 . 982,820 
 1 .011,261 
 
 1.028,885 
 
 1 . 040,769 
 1.049,019 
 1 . 055,020 
 1 . 059,524 
 
 1.744.382 
 
 1 . 372,080 
 1.246,808 
 1 . 188,968 
 1 . 157.328 
 
 1.138,161 
 I 125,443 
 
 1 . 1 16,705 
 1 . 1 10,400 
 
 1.105,686 
 
 1 . 000,000 
 1 . 062,500 
 1 .074,845,68 
 
 - 1.078,751.93 
 1.080,351,93 
 
 1 .081,123,534 
 1 .081,540,029 
 1 .081,784,170 
 1.081,936,586 
 1 . 082,039,586 
 
 00 
 
 1.082,323,233,7 
 
 1.082,323,233,7 
 
 1.082,323,233,7 
 
 . (88) 
 
68 
 
 The Atom 
 
 These are plotted as curves in Fig. 2. The value of 
 the constant h-,, as determined above in (59), is, therefore, 
 
 fer. 
 
 2a I 
 ttRx' 
 
 (92) 
 
 L 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 \, 
 
 
 
 
 
 — 
 
 "^ 
 
 
 - 
 
 y 
 
 X 
 
 s.^ 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 n 
 
 
 
 r 
 
 
 
 
 
 
 I 
 
 
 
 y 
 
 
 
 X 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ta 
 
 
 
 
 
 1.6 
 1.4 
 
 1.2 
 1.0 
 
 0.8 
 0.6 
 
 04 
 
 0.2 
 
 0.0 
 
 123456789 10 
 
 Fig. 2. 
 
 which is proportional to the reciprocal of the function 
 X shown in Fig. 2. The constant Btj, according to (61) 
 and (68), is 
 
 and is proportional to the function i +y/x. The func- 
 tions i/x, y/x, I +y/x and z/x are given in Table (94), 
 and shown as curves in Fig. 3. 
 
The Atom 
 
 69 
 
 T2 
 
 X 
 
 y 
 
 X 
 
 1+^ 
 
 X 
 
 Z 
 X 
 
 I 
 
 2 
 
 3 
 
 4 
 5 
 6 
 
 9 
 10 
 
 1.709,664,7 
 1 .201,616,9 
 1 .072,049,2 
 
 o!988!864!3 
 
 0-971,025,9 
 0.960,828,0 
 0.953,271,6 
 0.947,849,3 
 . 943,820,0 
 
 2.982,308,39 
 
 1.648,714,51 
 I 336,639,53 
 
 1.209,751.53 
 1 . 144,440,34 
 
 1 . 106,208,15 
 1.081,357,15 
 1.064,523,16 
 
 1.052,491,86 
 
 1.043,568,56 
 
 3.982,308,39 
 2.648,714,51 
 2.336,639,53 
 2.209,751,53 
 2.144,440,34 
 
 2.106,208,15 
 2.081,357,15 
 2,064,523,16 
 
 2.052,491,86 
 
 2 . 043,568,56 
 
 1.709,664,7 
 I .502,021,1 
 1.459,178,1 
 1.448,496,3 
 1.447.312,7 
 I .449,518,6 
 1.452,576,9 
 I .456,048,0 
 1.459,468,0 
 1 . 462,702,0 
 
 CO 
 
 0.923,938,4 
 
 1 . 000,000,00 
 
 2 . 000,000,00 
 
 1.519,818 
 
 (94) 
 
 3S 
 
 2-5 
 2.0 
 
 i.o 
 
 0.0 
 
 i 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 i\ 
 
 \, 
 
 
 
 
 
 
 
 
 \ 
 
 V 
 
 
 ~-^ 
 
 ,_^ 
 
 m 
 
 i4 
 
 
 
 A 
 
 V 
 
 
 z 
 
 X 
 
 
 nr 
 
 
 
 
 \ 
 
 ^ 
 
 
 ~~r~ 
 
 X 
 
 ___ 
 
 n 
 
 y 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 Tt 
 
 
 
 
 
 I 234.56789 10 
 
 Fig. 3- 
 
VIII 
 
 ET us next return to the initial conditions of 
 the two electrons expressed by the equations 
 from (35) to (43), for which purpose the 
 numerical values of the summations of v 
 and ju have been determined. Considering the initial 
 position vector (/o)o in (43), using the upper signs which 
 apply to the second electron, this is equivalent to 
 
 (P2)o = [i(fer,-BrJS(M+i')+a]i=[j(fer.-BrJ+a]i . (95) 
 
 We also have by (63) 
 
 - = -g- = ttK X 1.644,934, 
 and, by (92) and (93), 
 
 ferj — Btj 
 
 Hence 
 
 (p2)o = a\\ 
 
 a_/2 
 
 ~ -kK \x 2 X 
 
 _ 1.644,934 
 2 X 1.082,323 
 
 1.644,934 y 
 2 X 1.082,323 :v 
 
 1+^- 
 
 ' T — \ 
 
 I \ 
 
 1.082,323/ 
 
 + 
 
 2 X 1.644,934 
 
 which reduces to 
 
 (p2)o = a \ 0.240,091 + 3.289,868- 
 
 - 0.759.909 
 
 y 
 
 X 
 
 70 
 
 (96) 
 
 (97) 
 
 (98) 
 
 (99) 
 
The Atom 
 
 71 
 
 Again, we have the sum of the position vectors initially 
 in (37). 
 (P2 + Pi)o = -fer,S(i')i r>- '2i{v)i = za-i. . . (100) 
 
 2 TK X X 
 
 Hence, the value of (pi)o for the first electron is obtained 
 
 by subtracting (99) from (100), or the same value might 
 
 4..0 
 
 5-5 
 
 2.0 
 
 ••5 
 
 I.O 
 
 0.5 
 
 0.0 
 -0-5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 -~^ 
 
 
 _ 
 
 
 ^P/Pi 
 
 
 
 
 
 
 
 — 
 
 
 
 n 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . — 
 
 
 — 
 
 
 
 
 
 (P,)o 
 
 
 
 
 
 
 
 7-2 
 
 
 
 
 
 2 3 4. 5 6 7 
 
 Fig. 4. 
 
 10 
 
 be obtained from (43) by using the lower signs for the 
 first electron, namely 
 
72 
 
 The Atom 
 
 (pi)o = a I — 0.240,091 - 3.289,868 - 
 + 0.759,909^ + - 
 
 (lOl) 
 
 The following table (102) gives the values of (p2)o, (Pi)o, 
 (P2 + Pi)o and (p2 — pi)o initially for the first ten values 
 of the integer T2 and for n = «>. These functions are 
 plotted as curves in Fig. 4. From these we see that 62 
 is the outside electron and that it goes out to a maximum 
 distance from the nuclues of 3.598 radii of the original 
 and final orbit when T2 = i. It goes out to a minimum 
 distance 2.52 radii when T2 = «> . At the same time the 
 first electron is never initially so far away from the nucleus 
 as the final radius, its largest initial value corresponding 
 to the series when rj = 0° . 
 
 1-2 
 
 3.289,868^ 
 
 0.759.909^ 
 
 I 
 2 
 
 3 
 
 4 
 5 
 
 6 
 
 9 
 10 
 
 5.624,571,2 
 
 3.593,160,99 
 
 3.526,900,36 
 
 3-347.375.9 
 
 3-253.233 
 
 3.197.508 
 3.160,997 
 3- 136.138 
 3.118,299 
 3. 105,043 
 
 2 . 266,282,99 
 1 . 252,872,99 
 
 1.015,724,41 
 0.919,301,08 
 0.869,671 
 
 0.840,618 
 
 0.821,733 
 0.808,941 
 0.799.798 
 0.793.017 
 
 00 
 
 3-039.635.38 
 
 0.759,909 
 
The Atom 
 
 73 
 
 Tj 
 
 (P2)o 
 
 (P2 + Pl)o 
 
 (Pi)o 
 
 (P2 - Pl)o 
 
 I 
 
 2 
 
 3 
 
 4 
 S 
 
 3-598.379.2 
 2.940.379^ 
 2.751,266,6 
 2.668,156,8 
 2.623,654 
 
 3-419.329.4 
 3.004,042,2 
 2.918,356,2 
 2.896,992,6 
 2.894,625 
 
 -0.179,149,8 
 0.063,663,2 
 0.167,089,6 
 
 0.228,826,8 
 
 . 270,972 
 
 3.777.429.0 
 2.876,715.8 
 2.584,177,0 
 
 2-439.339.0 
 2.352,682 
 
 6 
 
 2.596,981 
 
 2.899,037 
 
 . 302,056 
 
 2 . 294,926 
 
 9 
 
 2-579.355 
 2.567,288 
 2-558.592 
 
 2.905,154 
 2.912,096 
 2.918,936 
 
 0-325.799 
 0.344,808 
 
 0.360,344 
 
 2.253.557 
 2 . 222,480 
 2.198,248 
 
 10 
 
 2.552.117 
 
 2.925,404 
 
 0-373.287 
 
 2.178,830 
 
 00 
 
 2.519,818 
 
 3 039.636 
 
 0.519,818 
 
 2 . 000,000,0 
 
 (102) 
 
IX 
 
 ET us next give some consideration to the 
 energy of the system composed of the hydro- 
 gen nucleus of charge + 2e and two negative 
 electrons. The system cannot be regarded as 
 a conservative system because it is capable of absorbing 
 and parting with energy from and to its external sur- 
 roundings. If the system were a conservative one, it 
 has been shown by Helmholtz in his paper " On the 
 conservation of force " in 1847 that the mutual forces 
 between any two material points must be in the line join- 
 ing them, and be a function of the distance between them. 
 The converse proposition is also true. If two material 
 points act on each other with a force depending as re- 
 gards magnitude on their mutual distance, but not in 
 the direction of the line joining them, they would be 
 capable of producing in each other an increasing velocity 
 and thus of generating or of dissipating energy. 
 
 The present electromagnetic theory as to the forces 
 between two electrical charges in motion makes their 
 directions not in the line joining their centers, and, ac- 
 cepting this as the fact, it follows that the system we 
 are now considering is not a conservative one, but one 
 capable of absorbing and of radiating energy. The fact 
 that these hydrogen atoms are known to radiate and to 
 absorb energy to and from external surroundings affords 
 a confirmation of the electromagnetic theory in respect to 
 the nature of the forces between two moving electrons, 
 namely that the forces do not at all times act in the line 
 joining their centers. 
 
 74 
 
The Atom 75 
 
 Let us denote the total energy content of the system, 
 say at the time zero, by Eq. This energy may be con- 
 sidered to be the sum of its kinetic energy. To, and a po- 
 tential energy, Vo, and we may write 
 
 To+V, = Eo (103) 
 
 At some later time, t, assuming that energy is being 
 diminished by radiation and lost to the system by an 
 amount, say Rt, since the time zero, the equation of energy 
 at this future time, t, is then 
 
 Ti + Vt = Eo-Rt, ... (104) 
 
 and, by subtraction of (104) from (103), the constant, 
 Eq, disappears, giving 
 
 {To - r.) + (Vo - VO = i?« (105) 
 
 If the time, /, is taken as infinite, meaning by this that 
 the two electrons have settled down in their final orbit 
 and have ceased to radiate energy, the equations which 
 have been discussed above tell us that the kinetic energy. 
 To, is equal to the final kinetic energy, T„. Writing the 
 energy equation (105) for the time t = 00, we then have 
 
 Vo - y« = i?„. . . . (106) 
 
 Now the energy radiated during any one excursion, 
 or rather during the return from this excursion of the two 
 electrons, will be different for each series, depending 
 upon the particular value of T2. But, since the electrons 
 come to the same final orbit every time, it is to be sup- 
 posed that the potential, V„, is a constant quantity for 
 all series. The initial potential energy, Vo, is, therefore, 
 equal to the energy radiated to within a constant, or 
 the difference between the initial potential energy and 
 the energy radiated is constant for all series. 
 
 It now seems entirely legitimate to apply the Einstein 
 
76 The Atom 
 
 equation for the total energy radiated to the problem. 
 This equation may be expressed as follows: 
 
 Energy radiated = hv' = —v, . . . . (107) 
 
 where v' denotes the frequency and v the angular velocity 
 of each component frequency emitted by the system 
 during the return of the electrons after displacement. 
 According to this theory these frequencies are infinite 
 in number corresponding to values of t from i to infinity, 
 T2 being a fixed integer for any one excursion. The 
 total energy radiated in one operation of the system is, 
 therefore, 
 
 R„ = — lv = hKI^\ = bKz. (See (72).). . (108) 
 
 27r I T 
 
 This energy is evidently a function of the initial posi- 
 tion vectors of the two electrons, for we have in (100) 
 
 (p2 + Pi)o = ^^p' (109) 
 
 and, denoting the scalar values of ps and pi by r2 and n, 
 we obtain the ratio of /?„ to r^ + n as follows: 
 
 R„ hK , . 
 
 — f- = — X (no) 
 
 Ti + n 2a 
 
 That is to say, this ratio is proportional to the function, 
 X, given in Table (88) ^and plotted in Fig. 2. It follows 
 from this that the initial potential, Vo, is also a function 
 of the initial sum of the distances because we have shown 
 above that it differs from R^hy a constant quantity. 
 We now have the equation, according to (106) and (108), 
 
 Vo-V„=i?„=^S»' (Ill) 
 
 or Vo Sf = V„, a constant. . . . (112) 
 
 27r 
 
The Atom 77 
 
 Let us next seek for some expression for Vo which will 
 reduce the difference, Vo~ R„, to a constant quantity, 
 
 the R„ = — Si* being an infinite series of terms. Evi- 
 27r 
 
 dently the value 
 
 y„ = - A S;x = _ hK(^^ - z), (See (75).). (113) 
 
 will accomplish this, for we have 
 
 h h TT^ 
 
 - — (Sm + Sj') = - — si = - hK- 
 
 2ir^ '^ ' 27r 3 
 
 = V„. (See (63).) (114) 
 
 Of course, an assumed value for Vo which differs from 
 (113) by a constant would also satisfy the condition in 
 (112), but it is apparent that the principal part of Vo 
 must contain a series similar to S/x, which will reduce 
 
 the difference between Vo and — Si' to a constant. There 
 
 2X 
 
 is no proof at present that no constant should be included, 
 except that the numerical values obtained by adhering 
 
 to the very simple expression for Vo, S/x, which is 
 
 analogous with the value of /?„ = — 'Zv, gives a result 
 
 in remarkable agreement with the experimental voltages 
 of ionization for hydrogen, as will presently be shown. 
 
 The energy, Y^, required to separate the two electrons 
 completely away from the nucleus if they start from 
 their original or final orbit is, by (114), 
 
 y„ = - ibK - = - 6.547 X 10-" X 3-290 X io>s X 3.289,868 
 
 = - .2154 X I0~^° — 
 
 = - .7086 X 1 0-1* ergs, (115) 
 
yS The Atom 
 
 and the energy per electron is one half of this or 
 - -3543 X lo"!" ergs. The energy required to separate 
 the two electrons at zero time when the electron, ej, is 
 already in its position of maximum distance from the 
 nucleus is given by Vo in (113). The following Table 
 (116) gives the values of Vo for the first ten values of 
 T2. This energy grows smaller and smaller with increas- 
 ing values of T2. When T2 = i, we have 
 
 V.= -4k(|-.) 
 
 = - 04932 X I o~'* ergs . . . . (117) 
 WhenT2 = 2, Vo = -hK(—~i.25\ 
 
 = - 0.4394 X 1 0-1' ergs . . . . (118) 
 When T2 = 00, Vo = _ bKf- -'^\ 
 
 = -0.3543 X 1 0-1" ergs . . . . (119) 
 
 T2 
 
 Vo^-hK(^^ 
 
 - 2) = - 0.2154 X lo-i'Cs. 289,868 
 
 -z) 
 
 I 
 
 - .2154 X 10-" X 2.289,868 = - 0.4932 X 
 
 lo-'" ergs 
 
 2 
 
 
 X 2.039,868 = - 0.4394 
 
 
 
 3 
 
 " 
 
 X 1.928,757 = -0.4155 
 
 
 
 4 
 
 
 X 1.866,257 = - 0.4020 
 
 
 
 5 
 
 
 X 1.826,257 = -0.3934 
 
 
 
 6 
 
 7 
 8 
 
 
 X 1.798.480= -0.3874 
 X 1.778,071 = -0.3830 
 X 1.762,446 = -0.3796 
 
 
 
 9 
 10 
 
 " 
 
 X 1.750,100 = -0.3770 
 X 1.740,100 = -0.3748 
 
 
 
 
 
 00 
 
 ** 
 
 X 1.644,934 = -0.3543 
 
 " 
 
 *t 
 
 (116) 
 
 The energy required per electron is one half of thesa 
 values. It is customary to express energy as the product 
 of electromotive force and charge for the reason that it is 
 
The Atom 
 
 79 
 
 electromotive force that is observed by those who have 
 conducted experiments to determine the critical values 
 at which ionization of the gas takes place. When an 
 electron of charge, e, is driven by an electromotive force, 
 E, the energy, T, required is 
 
 T 
 
 T = eE, and E=- (120) 
 
 e 
 
 If absolute units are used, ergs for energy, and absolute 
 electrostatic units for charge, we obtain the electromotive 
 force in absolute electrostatic units and not in volts. 
 To convert into the practical units of electromotive 
 force, volts, multiply by c X io~*, giving 
 
 Tc 
 E =— X 10-* volts, (121) 
 
 where T is expressed in ergs, and e = 4.774 X io~" 
 electrostatic units. Hence 
 
 £ = 0.6284 X I o'T volts (122) 
 
 By means of this formula we may find the voltage 
 required to impart to a single electron the energy T ergs. 
 Substituting for T the energy required per electron to 
 separate it from the atom, we obtain the following table 
 
 T2 
 
 E = ionizing voltage 
 
 I 
 
 2 
 
 3 
 
 4 
 5 
 6 
 7 
 8 
 
 9 
 10 
 
 00 
 
 13.806 
 13 055 
 12.631 
 12.361 
 
 12.172 
 12,034 
 11.927 
 11.845 
 11.776 
 
 II. 132 
 
 (123) 
 
8o The Atom 
 
 (123) of ionizing voltages, which may be compared with 
 the experimental values. 
 
 It is considered that this theoretical result is in re- 
 markable agreement with the recent experimental de- 
 termination of the ionizing voltages for hydrogen by 
 Davis and Goucher and others. These experimental re- 
 sults show that nothing whatever begins to happen in 
 hydrogen until 11 volts is passed, and at a value just a 
 little above 11 volts ionization sets in. According to 
 the Table (123) ionization begins at 11. 13 volts, corre- 
 sponding to a value of Tj equal to m . According to the 
 theory the ionization should be almost continuous from 
 1 1. 1 3 volts up until we come to the very small values of 
 T2, when the voltage takes larger jumps, ending with a 
 maximum value of 15.5 volts. Davis and Goucher have 
 observed a new type of ionization setting in at about 15.8 
 volts, which is only .3 of a volt greater than the theo- 
 retical maximum in the table. This small fraction of a 
 volt is probably within the error of experimental measure- 
 ment. Another type of ionization has been observed at 
 13.6 volts, which corresponds very closely with the second 
 value in the table corresponding to T2 = 2. All values 
 corresponding to higher values of T2 are too near together 
 to show as critical points in any of the experimental 
 curves since they are merged into one another. 
 
 It is of interest to contrast this result with the values 
 of the ionizing voltage for hydrogen derived from the 
 Bohr theory by the experimenters above referred to. 
 By the use of the formula 
 
 r = i.' = i^:(i,-±) (,24) 
 
 it is possible to derive, by putting Ts = i, and ti = 00, 
 T = bK, (125) 
 
The Atom 8i 
 
 and by putting Tj = i, and Ti = 2, to derive 
 
 T^^hK (126) 
 
 4 
 
 The bK in (125) converted into volts gives 
 
 E = X lo"' = 13.54 volts, . . . (127) 
 
 and the value in (126) is three quarters of this, namely 
 
 E = 10.15 volts (128) 
 
 The former value 13.54 is the largest value of the voltage 
 given by this formula, and the 10.15 is the smallest value 
 obtainable when Ta = i. If Ta were greater than unity, 
 it is possible to obtain as small values as we please way 
 down to zero. 
 
 The 10.15 volts is too small to represent any experi- 
 mental ionizing voltage by nearly one volt, since nothing 
 whatever happens until 1 1 volts are passed. The 13.54 
 falls near to the observed value of another type of ionizing 
 voltage, but the 13.54 is the maximum possible value 
 that this formula yields for any values of r whatever, 
 and there is no indication of any higher voltage of ioniza- 
 tion in hydrogen given by the Bohr theory. 
 
 It is difficult to see why the voltages calculated in this 
 way from the formula (124) should represent ionizing 
 voltages at all, unless we always put Ti = m. 
 

 M 
 
 1 
 
 |P to this point nothing has been learned con- 
 cerning the absolute value of the kinetic 
 energy. To, nor T„, which is equal to it, 
 since they canceled each other and dtopped 
 out from the energy equation (105). Neither have we 
 obtained any absolute value for the final radius of the 
 orbit, a, in centimeters. The Bohr theory made use 
 of the following very general theorem, upon which its 
 most important results are based: " In every system 
 consisting of electrons and positive nuclei, in which the 
 nuclei are at rest and the electrons move in circular 
 orbits with a velocity small compared with the velocity 
 of light, the kinetic energy will be numerically equal to 
 half the potential energy." If we should make use of 
 this theorem, the kinetic energy of the two electrons in 
 this system would be equal to one half of the potential 
 
 energy in (114), bK-r- The theorem quoted, however, 
 
 assumes the inverse square law of force between electrons, 
 which we have not assumed and do not propose to as- 
 sume. Another result from the same theorem applied 
 to the several stationary orbits in the Bohr theory was 
 that the energy radiated is equal to the change in the 
 kinetic energy, or that the sum of the energy radiated in 
 coming from an infinite distance to the nucleus and the 
 final kinetic energy is equal to the final potential energy. 
 In the theory expressed by the equations given above, 
 there is no change in the kinetic energy between the outer- 
 
 82 
 
The Atom 83 
 
 most position of the electron, d, and its final position. 
 This is in direct contradiction to the theorem quoted. 
 
 It will be noticed that the energy radiated as expressed 
 in (108) depends upon the value of T2, and that it varies 
 
 ■7^2 
 
 between the values hK, when Ti = 1, and hK-p when 
 
 o 
 
 Ta = 00. The greater of these two values ;s just equal 
 to half the final potential energy, as above shown, but 
 it corresponds to the series where T2 = 00. The energy 
 to separate the two electrons from the nucleus in this 
 series is a minimum (see (119)), while the energy to 
 separate them in the series where T2 = i is a maximum 
 (see (117)). We shall, therefore, regard the first series, 
 where T2 = i, as the fundamental series, and shall con- 
 sider that the kinetic energy of the two electrons in their 
 final orbit is equal to the energy radiated in this series 
 rather than in the head series, where T2 = 00. This 
 energy is simply hK. The justification of this is to be 
 found in the fact that it leads to the formula for the 
 velocity of electrons in rings in complete agreement with 
 a result previously obtained, and from which a correct 
 numerical value of the Newtonian gravitational constant 
 has been derived, as will be shown in a subsequent sec- 
 tion. Let us, therefore, equate the kinetic energy of 
 the two electrons in their final orbit to hK, giving 
 
 hK = i^mv^ = mi^. . .... (129) 
 
 From this is derived an expression for the velocity of 
 the electrons in the ring, for, in any circular orbit where 
 a point revolves with a frequency, n, the linear velocity 
 
 is 
 
 V = nran. (130) 
 
 But the frequency in the final orbit, as determined by 
 the above equations (see (52)) is 2K, hence 
 
84 The Atom 
 
 V = 4TrKa (131) 
 
 and v^ = ;8V = iGTT^K^a^ (132) 
 
 Eliminating v^ between (129) and (132), we have 
 
 hK = i6Tr^moK^a\ or a^ = ^^^^^j^ ' (i33) 
 Whence 
 
 ~ 47r \maKj 
 
 Substituting in this the known values of h, mo and K, 
 namely 
 
 h = 6.547 X 10-2', (135) 
 
 mo = 0.90 X 10-" (136) 
 
 and K = 3.290 X lo'*, (137) 
 
 we obtain numerically 
 
 a = .374 X io-%m., (138) 
 
 and by (131), 
 
 V = 1.542 X 10*, and j8 = 0.00514. . (139) 
 
 It has been found by Dr. Bohr that the value of K, 
 Rydberg's constant, may be expressed with a surprising 
 degree of accuracy in terms of the properties of the 
 electrons by the following formula: 
 
 K p— (140) 
 
 There are some coincidences as to the numerical value 
 of K that are worth mentioning. In the first place the 
 significant figures of K, 3.290, are very close indeed to 
 the value of ir'^/s = 3.289,868, which plays such a promi- 
 nent r6Ie in this theory. If we separate (140) into two 
 factors, thus, 
 
 ^ = - X -p-. (141) 
 
The Atom 85 
 
 it is seen that the value of 6moe*/b^ must be remarkably 
 close to the even number 10'*. Using the values in (135) 
 and (136), and putting 
 
 e= 4.774 X lo-i", (142) 
 
 6moe* 
 
 we obtain ^„ 4 
 
 ^3 - 0-99953 X ioi« (143) 
 
 So far as can be seen there is no theoretical support 
 in this theory for the combination of quantities ma, e 
 and h in (140). It seems rather as if the Rydberg con- 
 stant should be connected in some way with the properties 
 of the nucleus of the atom. We have already seen that 
 there is strong support for the Lorentz mass formula 
 given in (3) above and repeated here. 
 
 4/eV 4 fey , . 
 
 m=— - or = — -. ... (144) 
 
 Sa\cJ $m\cj ^ ' 
 
 If this formula is applied to the nucleus of the hydrogen 
 atom, of charge 2e, and the following values are used, 
 
 m„ = 1.662 X \o~^\ (145) 
 
 2e = 2 X 4.774 X 10"'", . . . (146) 
 
 c = 3 X iQi", (147) 
 
 we have _ 3-2 / 4.774 X 10-'° ^ 
 
 "^ 1.662 xio-^A 3x10"' / 
 
 = 4.8756 X io-"cm (148) 
 
 The reciprocal of the expression — (-j , which occurs 
 
 in this formula, is numerically nearly equal to K. If 
 it is taken to be exactly equal numerically, we have a 
 new relation 
 
 2K = mH\^^ (149) 
 
 and mu = 2K[--\ (150) 
 
86 The Atom 
 
 Substituting in this the above value of e, and the value 
 K = 3.290 X 10^', we obtain numerically 
 
 mj] = 1.666 X io~2* (151) 
 
 If, however, the value of K of (149) is substituted in 
 the expression for the radius (144), the ma, e and c dis- 
 appear, and we obtain the simple expression connecting 
 the radius of the nucleus with K as follows : 
 
 °^=^ ('^^^ 
 
 This gives numerically a slightly smaller radius than 
 was obtained in (148) above, namely, 
 
 an = 4.8620 X lo""^' cm (153) 
 
 Let us assume that this is the correct value of the 
 radius because it is derived from a theoretical relation 
 involving but one constant, K, which is known with pre- 
 cision. This gives an exact value for e^/mn as is evident 
 by solving both (144) and (149) for this quantity, giving 
 
 £ =^''«^'=^= '-36778 xio«. . . . (154) 
 
 The decimal places are retained, although they have no 
 particular meaning beyond the fourth significant figure, 
 because we have to separate this result into the two 
 factors, e and iuh. 
 
 There is an apparent difficulty here, however, in adopt- 
 ing the relation between Ca and K in (152) because it 
 makes K have the dimensions of an inverse length ap- 
 parently. The dimensions of K should be those of a 
 frequency according to (22) above, that is to say, K 
 should have the dimension T~'. 
 
 It should be pointed out that the difficulty lies rather 
 with the Lorentz formula (144), for the specific inductive 
 
The Atom 87 
 
 capacity of the medium, k, has been suppressed. Ac- 
 cording to the electrostatic system of units, we have 
 the following dimensions: 
 
 e = MiDT-^ki, (156) 
 
 c = LT-\ (157) 
 
 e/c = MiDki, . . . . (158) 
 
 ,4^^/^)^ = ^^ (^39) 
 
 This is the quantity that appears on the right of (144), 
 and on the left appears simply a, which has the dimension 
 L, and not Lk. Or, in other words, the k is entirely 
 ignored in this equation, and it is assumed that it is 
 dimensionless. If we do not admit that this is so, and 
 attribute to k some dimensions in terms of L and T, we 
 may, by arbitrarily giving to k the dimensions of the re- 
 ciprocal of a velocity, namely L''^T, correct the apparent 
 difficulty. The corrected equation is as follows : 
 
 «-^=i;(?T ('60) 
 
 The dimensions of both sides of the equation now agree, 
 and are equal to LLr^T = T. 
 
 In a similar manner the k should be included with equa- 
 tion (152), making it 
 
 aak = 8/5K (161) 
 
 This makes the Rydberg constant, K, the reciprocal of 
 a time, T~^, as it should be. So long as k has a unit 
 value, this change will make no difference in the numerical 
 resultsralready given. 
 
 It seems worth remarking before concluding the dis- 
 cussion of the new expression for K in (149) that, if we 
 express the single charge, e, in electromagnetic units, in- 
 
88 The Atom 
 
 stead of electrostatic units, the velocity of light disappears, 
 leaving the simple relation, 
 
 2i^ = -J- electromagnetic units. . . . (162) 
 
 We have seen that the value of the Rydberg constant 
 adopted in (149) gives an exact value of the ratio e^/mn, 
 but it gives neither quantity individually. There is an- 
 other experimental equation derived from experiments 
 on the electrochemical equivalent of silver as follows: 
 
 ;zr = 96494. (163) 
 
 emu 
 
 the constant 9649.4 sometimes being called the Faraday 
 constant. The Ag denotes the atomic weight of hydro- 
 gen referred to oxygen =16. This equation gives an 
 independent value of the ratio of e/mg. By dividing the 
 ratio e^/niH in (154) by the ratio e/ms in (163) the m^ 
 cancels, and the following value of e is obtained: 
 
 e = 4.763 X lo-i" (164) 
 
 And using this value of e in (163) the value of ma 
 becomes, 
 
 ma = 1.658 X io~^* (165) 
 
 It is considered that these numerical values are both 
 within the experimental error of determining them. 
 
 By the use of another experimental result, namely, 
 the determination of the ratio of the charge to the mass 
 of the electron, a value for the mass of the electron is 
 determined. We have 
 
 ^^=1.767x10' (166) 
 
 The experimental constant 1.767 X 10' is that de- 
 termined by Bucherer, and it may be referred to as 
 
The Atom 89 
 
 Bucherer's constant. In order to obtain a system of 
 values that is consistent throughout with the theo- 
 retical formulae adopted, we shall use the value of e 
 as determined by these formulae in (164), namely, 
 4.763 X iQ-i", giving 
 
 mo = .898 X 10"*' grams (167) 
 
XI 
 
 ET us now return to the consideration of the 
 velocity of the electrons in a ring of elec- 
 trons as given by (129) above. If the value 
 of K in (149) is substituted for the K in 
 
 (129), we find 
 
 and 
 
 1.2= - 
 
 hnifi 
 2 mo 
 
 (168) 
 
 whence 
 
 and 
 
 
 = 
 
 2e 
 hmii 
 2e^mo ' 
 
 'bm,H 
 ma 
 
 (170) 
 
 (17O 
 
 ^ 26 y mo ^ ' ' 
 
 In this case the formula represents the velocity of 
 an electron in a ring o two electrons, the 2 under the 
 radical representing the number of electrons in the ring. 
 If the 2 is replaced by p, as representing the number of 
 electrons in any ring, the formula becomes 
 
 _ Vp c /bmjs ^ Ve 
 
 2 ej/ mo 
 
 The chief characteristic of this formula is that it makes 
 the velocity of the electrons in the ring independent of 
 the radius of the ring. It is considered that this is ap- 
 proximately true of any of the rings of electrons in atoms 
 that have many rings. The speed of any ring is de- 
 pendent only upon the number of electrons that it con- 
 tains and is independent of the velocities that other rings 
 
 90 
 
The Atom 91 
 
 in the same atom may have. It is considered also that 
 this formula is not exact but very approximate, and 
 that the true velocity is dependent upon the radius but 
 only to the second order. The reason for holding this 
 view will be given in a subsequent section, where it is 
 also pointed out how great the variation from this formula 
 probably is. 
 
 This formula, and the dependency of the velocity of 
 the electrons in a ring merely upon the number of electrons 
 in the ring, is a radical departure from previous theories 
 of the atom. It contains within it the idea that the 
 cause of the revolution of the ring is the mutual action 
 of the electrons in the ring upon each other. Electro- 
 magnetic theory shows that the force exerted upon a 
 single electron in the ring by all of the other electrons 
 always has a positive component along the tangent to 
 the orbit in the direction of the motion. This tan- 
 gential force must be reduced to zero by some equal and 
 opposite force before there can be a steady and uniform 
 motion of the ring. The force that is supposed to counter- 
 balance this positive force along the tangent line is the 
 tangential reaction of the electron being considered upon 
 itself. The only way that this problem can be treated 
 by electromagnetic theory is to make certain hypotheses 
 concerning the electron itself, such, for example, as to 
 assume that we have the solid Lorentz electron or some 
 other form that has been proposed. The forces that are 
 derived from these different forms of hypotheses differ 
 according to the hypothesis, and the problem can have 
 no certain and definite solution. The hypotheses that 
 have to be made really beg the question. 
 
 It is not, therefore, sufficient to say that there is 
 complete equilibrium and uniform velocity of the ring if 
 we merely equate the forces normal to the orbit, or along 
 
92 The Atom 
 
 the radius, to zero. The particular velocity that this 
 would result in may not be such as to cause the tangential 
 forces of the other electrons in the ring to balance the 
 force that the electron exerts upon itself at this velocity. 
 If so, these tangential forces will alter the velocity until 
 they do balance, and the radius will have to change 
 accordingly until the radial forces also balance. The 
 number of electrons in the ring may in this way control 
 the speed. 
 
 There has always existed a difficulty in considering 
 rings of electrons from the point of view of electromagnetic 
 theory, for this theory shows that there is of necessity 
 a certain amount of radiation of energy from a ring of 
 electrons unless the number of electrons is very large. 
 For example, if it is assumed that the rate of radiation of 
 energy from a single electron in an orbit is unity according 
 to the theory, then the rate of radiation from a ring of 
 two electrons is about 4000 times smaller, and from a 
 ring of three about forty million times smaller, and from 
 a ring of four about a million million times smaller, and so 
 on, the rate falling off with very great rapidity for a small 
 increase in the number of electrons. In looking at this 
 matter from the standpoint of the equations above given, 
 and the theory as above outlined, it is seen that the final 
 orbit really corresponds to the case where Ta = in (24). 
 Putting Ta = o in this we have 
 
 " = 2^^(72-72) =0 (173) 
 
 and there is no frequency of radiation. The same result 
 is obtained if any one of the series of frequencies is 
 examined, which, as we have seen, always end with a 
 zero frequency when the final orbit is attained. It is 
 certainly stretching the logic of the case to say that there 
 
The Atom 93 
 
 is no radiation of energy simply because the frequency of 
 the radiation has been reduced to zero. By analogy, 
 when the frequency of an alternating current is reduced 
 to zero, the result is a steady current and not a zero 
 current. And by further analogy, an alternating-current 
 instrument may be conceived that will show no record 
 of a direct current, although this is not a common form 
 of these instruments. It may easily be imagined that the 
 photographic plate and our eyes are such instruments, as 
 regards the energy radiated, as will show nothing when 
 the frequency is reduced too low. This would make the 
 apparent energy radiated, as expressed by (108), reduce to 
 zero when the frequency falls to zero in accordance with 
 observations. The actual direct-current energy may still 
 be present and escape all observation. It is exceedingly 
 small anyway, if we may trust electromagnetic theory 
 for its value. A revised form of electromagnetic theory, 
 and there is little doubt that it will eventually be revised, 
 seems likely to make this theoretical energy of radiation 
 smaller than the present theory does. The reasons for 
 holding this view will be given in a later section. There 
 is no necessity, so far as can be seen, to make the energy 
 radiated from a ring of electrons in the final steady orbit 
 exactly zero except possibly the difficulty experienced in 
 accounting for the source of the energy. The rate of 
 radiation is probably so slow that the internal energy 
 of the electron itself would be capable of sustaining it 
 for lengths of time so great that it has been as yet im- 
 possible to detect any change. And again the change, 
 when it comes, may be of the nature of a sudden change 
 analogous to the sudden alterations in the atoms as 
 they disintegrate, which would escape observation. This 
 matter of the source of the energy has been seriously 
 considered by writers on electromagnetic theory, but, so 
 
94 The Atom 
 
 far as any definite results are concerned they may be 
 considered to be negligible. It is a pure speculation to 
 imagine anything about it in the present state of our 
 knowledge, but this account would have been deficient 
 had all reference to these difficulties been omitted. 
 
 Let us next return to the formula for the radius of 
 the orbit in (134). This may also be expressed in terms 
 of the properties of the electrons without the quantity 
 K. If the value of K of (167) is substituted in (134), 
 we have 
 
 a = = .374 X 1 0-* cm. . . . (174) 
 
 4X c\momH/ 
 
 This radius .374 X io~* for the ring of two in hydrogen 
 is smaller than the smallest orbit for the single electron 
 of the Bohr theory, which is about .529 x lo"*, the next 
 orbit being 4 times greater. 
 
 The absolute value of the smallest orbit in hydrogen 
 is of considerable interest and importance. A value so 
 large as .529 x io~* has presented considerable difficulty, 
 especially when there is but a single electron in an orbit, 
 for it is possible to liquefy hydrogen, and in this state 
 the average distance between the centers of the atoms 
 may be calculated with considerable certainty from the 
 density of the liquid. This distance is of the same order as 
 the distances between atoms in crystals of various kinds, 
 say between 2 and 3 x io~* centimeters. If the radius 
 of the orbit of the electron can never be less than 
 .529 X io~* cm., then its diameter is 1.058 X io~*cm., 
 nearly half the distance between the centers of the atoms, 
 assuming that this distance is 2 x lO"* cm., or one third, 
 assuming that the distance is 3 x lo"* cm. When the 
 electrons in adjacent atoms are at the nearest points of 
 their orbits there would be considerable interference, due 
 
The Atom 95 
 
 to their mutual action upon, each other, and this must 
 be so great that the system cannot be regarded as stable. 
 This difficulty is greatly reduced by reducing the size 
 of the radius about 30 %, but it is reduced very much 
 more by having two instead of one electron in the orbit. 
 The energy required to produce a disturbance in the orbit 
 must for some cause, as yet not known, rise above a 
 certain minimum value before radiation sets in, and, 
 provided this limit is not reached, there will be no radia- 
 tion. The hydrogen atom above described seems to be 
 superior in this respect to the single electron atom. 
 
XII 
 
 E will next compute the orbits of the two 
 electrons according to the equations, which 
 have been given above, in order that a 
 definite picture of the motion of the electrons 
 in certain instances may be obtained. For this purpose 
 let us first select the case of the " head" series, where 
 T2 = oo, as being the simplest for computation, although 
 there is considerable labor involved in obtaining any 
 numerical curve because it is expressed as an infinite 
 series of terms. 
 The energy radiated in this series is a maximum ac- 
 
 cording to (io8), being equal to hkz = hK -7 = .2154 
 
 X lo-i" X 1.644,934 = 0.3543 X lo-"* ergs. 
 
 The potential energy, Vo, required to separate the two 
 electrons completely away from the nucleus is a mini- 
 mum in this case, and it happens to be equal to hKz, 
 the same value as the radiated energy. 
 
 In this series, where Tj = <», we have 
 
 H^v = 2tK^ (175) 
 
 I T 
 
 and Sm = 2j' = 2tKS \ = - K (176) 
 
 27r 
 
 also S(jLi + J*) = 2Lv = — K. (177) 
 
 96 
 
The Atom 97 
 
 Let us first calculate the curve for the sum of the 
 radii pa + Pi as given in (27) above. The constant 
 
 multiplier is—', which is given by (59). In this the 
 
 2 
 
 Sy^ corresponding to T2 = 00, is given in the table (88). 
 Hence, in this series the constant multiplier is equal to 
 a/i.o82,3237r/C and the complete equation we have to 
 compute is 
 
 P2 + P1 
 a 
 
 i.o82,323ir/C 
 
 S I per'' (cos vt)i - j'e-'"(sin vt)j \ . . (178) 
 
 The initial value, when t = o, is given by (37) above 
 and again by (100). Using the values of x and z in the 
 tables (88) and (77) corresponding to T2 = «>, we have 
 the initial value 
 
 (P2 + Pi)o = 2a^i = 2a^^^g||| i = 3.039,63601'. . (179) 
 
 To obtain the first portion of the curve, near the time 
 f = o, we may develop e'"' cos vt and e""' sin vt in series 
 of powers of vt, giving the following values up to and 
 including (vty. This series development is required to 
 facilitate the addition of the infinite number of sines and 
 cosines expressed by the summation in (178), for this 
 reduces the process to finding the sums of the powers 
 of vt instead of the sines and cosines of these angles which 
 differ for every value of t. 
 
98 
 
 The Atom 
 
 e~'''sin vt = 
 
 + I 
 - I 
 
 I, I 
 
 [ill, l3li 
 
 l \i i3\3 Ell 
 
 L [Hi lili IJl 
 
 T"l7 + (i]i"51i+l6l7 
 
 + 
 + 
 
 + [etc 
 
 L Ii|9 + l3l7 l5l5'^Fl |9liJ 
 
 Jd'O' ■ 
 {my 
 
 {vty ■■ 
 
 {ny- 
 
 [vtY- 
 
 {vty - 
 
 {ny - 
 
 {vty - 
 
 (vtyo- 
 
 > + 1. 000,000,000,0 (v/) 
 ■■ - 1. 000,000,000,0 (j't)^ 
 
 ■■ + 0.333,333.333.3(1")' 
 
 o.ooo,ooo,ooo,o(vt)* 
 
 - 0.033,333,333,3 («)6 
 
 + 0.011,1 1 1,1 1 1,1 (j;t)' 
 
 - o.ooi,587,3oi,6(vt)' 
 o.ooo,ooo,ooo,o(vt)' 
 
 + 0.000,044,09 1, 6(ct)' 
 
 - 0.000,008,817,3(^0"' 
 (180) 
 
 e""' cos vt = 
 
 C+i 
 + C-I 
 
 2 
 
 I I 
 ill 11, 
 
 I I.I 
 I 
 
 + 
 
 +Letc, 
 
 6 + [ik."l^+ll 
 
 Ji L+^_ ' 
 
 il6. l3ll lllf. (7 
 I II I I 
 
 Jvty ■■ 
 {vty - 
 
 {vty. 
 
 {vty ^ 
 
 {.vty- 
 
 (vt)' ■■ 
 
 (vty- 
 (vty -- 
 
 + 1.000,000,000,0 
 
 - 1. 000,000,000,0 (j'O 
 
 0.000,000,000,0 {vty 
 
 + o.333,333,333,3(j;0» 
 
 : -o.i66,666,666,7(w)< 
 
 + 0.033,333,333,3 (i;f)6 
 
 0.000,000,000,0(1/0' 
 
 - o.ooi,587,3oi,6(»'0' 
 + 0.000,396,825,4(1/0* 
 
 - 0.000,044,091,6(110' 
 
 0.000,000,000,0 (j'O " 
 (181) 
 
The Atom 99 
 
 We have to multiply each term in these series by v, 
 this being a factor in (178), and this raises the powers 
 of V in each term by unity. Since, by (175), in this 
 series 
 
 SM = (27rK)"2^ (182) 
 
 the sum of all the terms in this series to infinity may be 
 found by the use of the table (67). Taking out the 
 factor 2irK, which appears in every term, we obtain 
 
100 
 
 The Atom 
 
 p2 + Pi. = 3a 1 1 .082,323 { 1 .000,000,000 S -^ 
 - 1.000,000,000 2 -J 
 
 0.000,000,000 "Z—. 
 
 0.333.333.333 2^ 
 -0.166,666,667 S^„ 
 
 0.033,333,333 S^ 
 
 0.000,000,000 S-n 
 
 - 0.001,587,3 S^ 
 0.000,396,825,42^ 
 
 - o.ooo,044,09i,62-2j 
 o.ooo,ooo,ooo,oS-g 
 
 + 20/1.082,323 {- 1.000,000,000 
 
 ^h^ 
 
 27ri<:0' 
 
 1.000,000,000 
 
 ^^< 
 
 « \2 
 
 - 0.333.333.333 
 
 ^,^< 
 
 
 0.000,000,000 
 
 S- ( 
 
 
 0.033,333,333 
 
 ^i' 
 
 
 - 0.011,111,111 
 
 ^,4,' 
 
 
 0.001,586,3 
 
 S- ( 
 
 
 0.000,000,000 
 
 S- ( 
 
 
 - 0.000,044,091, 
 
 62A.< 
 
 
 0.000,009,817, 
 
 3^( 
 
 " )10 
 
 ;2irii^0° 
 
 " )' 
 
 " )' 
 
 " )* 
 
 " )' 
 
 " y 
 
 " )' 
 
 " )' 
 
 li ■ (183) 
 
The Atom 
 
 lOI 
 
 Multiplying in the values of S — according to the table 
 (67) gives the following: 
 
 Pa + Pi = 20/ 1 .082,323 { 1 .644,934 (211 
 
 ■Kt)o 
 
 - 1.082,323 ( 
 
 " y 
 
 0.000,000 ( 
 
 ■ y 
 
 0.334,692 ( 
 
 ' y 
 
 - 0.166,832,5 ( 
 
 ■ y 
 
 0.033,415.3 ( 
 
 ' y 
 
 0.000,000,0 ( 
 
 • y 
 
 - 0.001,587,32 ( ' 
 
 ' y 
 
 0.000,396,826,9 ( 
 
 ' y 
 
 - 0.000,044,916,4 ( ' 
 
 ' y 
 
 0.000,000,000,0 ( 
 
 ' yo 
 
 + 20/1.082,323 {- 1.082,323 (2ir 
 
 Kty 
 
 1.017,343 ( ' 
 
 • y 
 
 - 0.334,692 ( 
 
 ' y 
 
 0.000,000 ( 
 
 ' y 
 
 0.033,415,3 ( 
 
 • y 
 
 -0.011,111,8 ( 
 
 ' y 
 
 0.001,587,32 ( 
 
 ' y 
 
 0.000,000,00 ( 
 
 ■ y 
 
 - 0.000,044,916,4 ( 
 
 ' )' 
 
 0.000,008,817,3 ( 
 
 ' )10 
 
 ]J 
 
 (184) 
 
 From this it is seen that, when f = 0, P2 + Pi 
 = 3.039,63601, but that the series is not useful beyond a 
 certain value of t. The highest term of the series has a 
 frequency K, and a period i/K. If t = i/K, or Kt = i, 
 this one component has executed one revolution. If we 
 set t = xjz-kK, or Kt = i/27r, the highest frequency term 
 has passed through i/2xth of a revolution, or about 57°. 
 The curve may be calculated up to this point of time 
 without sensible error due to the convergency of the 
 series. Letting 2irK( take in succession the values of 
 .1, .2, .3, etc., up to i.o, the following numerical values are 
 obtained. 
 
102 
 
 The Atom 
 
 
 (P2 + Pi)/ - 
 
 2a 
 082,323 
 
 (P2 + Pi)/a 
 
 iirKt 
 
 I 
 
 J 
 
 i 
 
 J 
 
 .0 
 
 1.644,934 
 
 . 000,000 
 
 3 039,636 
 
 0.000,000 
 
 .1 
 
 I 537,020 
 
 - 0.068,294 
 
 2 . 840,224 
 
 - 0. 126,198 
 
 .2 
 
 1.431,004 
 
 - 0.178,342 
 
 2.644,319 
 
 -0.329,554 
 
 •3 
 
 1 . 328,003 
 
 -0.242,099 
 
 2.453.986 
 
 - 0.447,369 
 
 .4 
 ■5 
 
 1.299,494 
 1.136,215 
 
 - 0.291,275 
 
 -0.327,779 
 
 2.271,953 
 2.099,585 
 
 -0.538,240 
 - . 605,695 
 
 .6 
 
 :8^ 
 
 1.048,718 
 0.967,558 
 0.892,781 
 
 -0,353,320 
 -0.369,483 
 
 -0.377,757 
 
 1.937,902 
 1.787,928 
 1.649,747 
 
 - 0.652,892 
 
 - . 682,759 
 -0.698,048 
 
 •9 
 
 I.Q 
 
 0.824,501 
 0.762,650 
 
 -0.379,471 
 -0.375.817 
 
 I 523.576 
 1.409,284 
 
 - 0.701,216 
 
 -0.694.464 (j8^) 
 
 Charting the equation (184) in this way not only gives 
 a curved path but locates points upon the curve at equal 
 intervals of time, from which the velocity of the point in 
 the curve is at once apparent. The curve is shown as 
 curve I, Fig. 5. 
 
 Let us next calculate the value of pi, from which, and 
 the value of pz + pi already found, we may find p2 and 
 P2 - pi graphically without the necessity for arithmeti- 
 cal calculation. Using the lower signs in equation (34) 
 we obtain 2pi; but it may be remarked that in the special 
 case we are now considering, that of the head series, 
 where v = ju, the first two or fexj-terms exactly cancel, 
 leaving only the B^j and A-terms. The value of Br, is 
 now, by (61) 
 
 SwKa/ y\ 
 
 Br 
 
 where nowy = x, and S2 = UttK)^ X 1.082,323. Whence 
 Br, = a/i.o82,3237rK' (187) 
 
The Atom 
 
 103 
 
 which is the same as — ', the coefficient in (184) above. 
 
 Hence 
 
 Pi = 
 
 S I j'e-=*"[(cos 2Vt)i - (sin 2vt)i\ 
 
 2 X i.o82,3237ri^ 
 — a[_{cos 2Pt)i - (sin 2Vt)j'] (188) 
 
 The values of e~'"sinj't and e""' cos ^'f given in (180) 
 and (181) above will also answer for this, if 2v is substi- 
 
 Fig. 5. 
 
 tuted in the place of v. Besides these are required for the 
 last term the expansion of sin 2vt and cos 2vt, which 
 follow. 
 
104 The Atom 
 
 sin (fi + v)t = sin 2vt 
 
 = I . 000,000,0 
 
 {iitKty 
 
 -0.166,666,6 
 
 ( " y 
 
 0.008,333,3 
 
 ( " Y 
 
 - 0.000,198,4 
 
 ( " )' 
 
 . 000,002,756 
 
 ( " )» 
 
 - . 000,000,025,052 
 
 ( " )" 
 
 = 1 . 000,000,0 
 
 {AtTrKt)" 
 
 - . 500,000,0 
 
 ( " y 
 
 0.041,666,6 
 
 ( " y 
 
 -0.001,388,8 
 
 ( " )• 
 
 0.000,024,802 
 
 ( " y 
 
 - 0.000,000,275,57 
 
 ( " r 
 
 . 000,000,002,088 
 
 ( " )« 
 
 (190) 
 
 COS {p, + v)t = COS 2Vt ■■ 
 
 (191) 
 
 By means of the tables (180), (181), (190) and (191) 
 we obtain the development of pi as follows: 
 
 a 
 
 ( 1.000,000 
 
 si- {2irKty 
 
 1.082,323 
 
 
 - 1.000,000 
 
 x:» 2^^( 
 
 " y 
 
 
 0.000,000 
 
 
 
 
 0.333.333 
 
 X8 2^( 
 
 " y 
 
 
 - 0.166,666 
 
 X 16 2^0 ( 
 
 •' y 
 
 
 0.033,333 
 
 X 32 2 i-, ( 
 
 " y 
 
 
 0.000,000 
 
 
 
 
 - 0.001,587,3 
 
 X "82^, ( 
 
 " )' 
 
 
 0.000,396,825,. 
 
 tx 2562^4-, ( 
 
 " )" 
 
 
 - 0.000,044,091,6 X 5i2S^ ( 
 
 " )' 
 
 
 0.000,000,000,0 
 
 
 It 
 
The Atom 
 
 105 
 
 ?_/ 
 
 1.082.323 I 
 
 1.000,000 
 - 1.000,000 
 
 0.333.333 
 
 0.000,000 
 
 - 0.033,333 
 
 0.011,111 
 
 X2 i:~ iiwKty 
 
 X 8 si- ( " )» 
 
 X32S^, ( " )» 
 x64Si-, ( " )' 
 X i28S^, ( " )' 
 
 - 0.001,587,3 
 0.000,000,0 
 
 0.000,044,091,6 X 5 "2S^„ ( " Y 
 
 - 0.000,008,817,3x10242-^ ( " )ii 
 
 M 
 
 {2irKt)o 
 
 — 0.500,000 x 4 
 
 
 ( " y 
 
 0.041,666 x 16 
 
 
 ( " y 
 
 - 0.001,388 X 64 
 
 
 ( " )' 
 
 0.000,024,801,625 X 
 
 256 
 
 ( " y 
 
 - 0.000,000,275,573 X 
 
 1024 
 
 ( " y 
 
 0.000,000,002,088 X 
 
 4096 
 
 ( " )" 
 
 + a{ 1.000,000X2 
 
 
 (,2irKty 
 
 - 0.166,666 X 8 
 
 
 ( " y 
 
 0.008,333 X 32 
 
 
 ( " )= 
 
 - 0.000,198,413 X 128 
 
 
 ( " y 
 
 0.000,002,755,73 X 5 
 
 12 
 
 ( " y 
 
 — 0.000,000,025,052 X 2048 ( " )'i 
 
 !/■ (192) 
 
 Multiplying in the numerical values of S — „ S -;, etc., 
 
 T T 
 
 according to the table (67), and the constant 1/1.082,323, 
 we finally obtain 
 
io6 
 
 The Atom 
 
 pi 
 
 = a{ 1.519,817,75 
 
 (2ta:o° 
 
 - 2.000,000,00 
 
 
 " y 
 
 0.000,000,00 
 
 
 ti y 
 
 2.473,881,668,7 
 
 
 ti \3 
 
 - 2.466,286,196,6 
 
 
 •' .)4 
 
 0.985,776,820,1 
 
 
 ' y 
 
 0.000,000,000,0 
 
 
 ' y 
 
 - 0.187,723,499 
 
 
 ' y 
 
 0.093,860,767,9 
 
 
 ' y 
 
 - 0.020,857,836,2 
 
 
 ' y 
 
 0.000,000,000,0 
 
 " yo 
 
 + a{- 2.000,000,000,0 
 
 2TKty 
 
 3.759.848,283,8 
 
 
 ' y 
 
 - 2.473,881,668,7 
 
 
 ' y 
 
 0.000,000,000,0 
 
 
 ' y 
 
 0.985,776,820,1 
 
 
 ' y 
 
 - 0.657,042,959,4 
 
 [ 
 
 ' y 
 
 0.187,723,499 
 
 
 ' V 
 
 0.000,000,000 
 
 
 ' y 
 
 - 0.020,857,836,2 
 
 
 ' y 
 
 0.008,342,163,5 ( 
 
 " y° 
 
 + a{- 1.000,000,000 
 
 2irKt)o 
 
 2.000,000,000 
 
 « y 
 
 - 0.666,666,666 ( 
 
 .. y 
 
 0.088,888,888 
 
 " )6 
 
 - 0.006,349,216 ( 
 
 '■ r 
 
 0.000,282,187 ( 
 
 " )10 
 
 + a{ 2.000,000,000 ( 
 
 zirKty 
 
 - I -333,333,333 ( 
 
 " y 
 
 0.266,666,666 ( 
 
 " y 
 
 - 0.025,396,86 ( 
 
 " y 
 
 0.001,410,93 ( 
 
 I 
 
 y 
 
 ]j 
 
 (193) 
 
 Adding the two sets of i and j terms in the above, we 
 obtain 
 
The Atom 
 
 107 
 
 Oi = a\ 
 
 + "( 
 
 0.519,817,75 (2T 
 
 Kt)<> 
 
 • 2.000,000,00 ( ' 
 
 ' y 
 
 2.000,000,00 ( 
 
 ' r 
 
 2.473,881,668,7 ( 
 
 ' y 
 
 3.132,952,863,3 ( 
 
 ' y 
 
 0.985,776,820,1 ( 
 
 ' y 
 
 0.088,888,888,8 ( 
 
 ' y 
 
 0.187,723,499 ( ' 
 
 ' y 
 
 0.087,511,551,9 ( 
 
 ' y 
 
 - 0.020,857,836,2 ( 
 
 • y 
 
 0.000,282,187 ( 
 
 ' r 
 
 0.000,000,000,0 (2Tr 
 
 Kty 
 
 3.759,849,283,8 ( 
 
 • y 
 
 - 3.807,215,002,0 ( 
 
 ' y 
 
 0.000,000,000,0 ( 
 
 • y 
 
 1.252,443,486,8 ( 
 
 ' y 
 
 - 0.657,042,959,4 ( 
 
 ' y 
 
 0.162,326,639 ( 
 
 ' y 
 
 0.000,000,000 ( 
 
 ' y 
 
 - 0.019,446,906,2 ( 
 
 ' y 
 
 0.008,342,163,5 ( 
 
 ' y 
 
 !i 
 
 (194) 
 
 From this the following table of numerical values of 
 (pi /o) have been obtained for a series of values of 2TrKt, 
 namely, .0, .1, .2, .3, etc., up to 1.0. 
 
 2irKt 
 
 Pi/a 
 
 i 
 
 J 
 
 .0 
 
 . I 
 .2 
 •3 
 
 • 4 
 ■5 
 
 .6 
 
 •9 
 1 .0 
 
 0.519,818 
 0.341,988 
 0.214,627 
 0. 143,660 
 0. 128,122 
 0. 164,411 
 
 244,953 
 0.361,026 
 0.502,053 
 0.656,972 
 0.814,625 
 
 . 000,000 
 0.033,803 
 0. 120,297 
 0.238,192 
 0.368,310 
 495.533 
 
 0.602,321 
 0.682,852 
 0.727,498 
 733,409 
 699,257 
 
 (195) 
 
io8 The Atom 
 
 The curve obtained from this table is charted as curve 
 II in Fig. 5. It represents the path followed by the 
 first electron from the moment when the radiation of 
 energy begins up to a time t = i/iirK. Since we already 
 have the sum of pi and pi in curve I, the path of the 
 second electron is obtained by subtracting pi from pi + pi. 
 The position vector of ej is then equal to the line joining 
 a given point in curve II with the corresponding point 
 in curve I at the same time. These vectors, when trans- 
 ferred to the origin, or the nucleus of the atom, give 
 the path of d as curve III. The difference between p2 
 and pi is the vector from a given point in curve II to the 
 corresponding point of curve III, and this gives the curve 
 IV, Fig. 5. 
 
 It may be seen from these curves that the paths of 
 both ci and 62 are approaching the small circular orbit, 
 the full line, and will eventually arrive at the opposite 
 ends of a common diameter of this orbit. The curve I, 
 representing the sum of the position vectors, approaches 
 the origin, finally becoming zero when radii of ei and 62 
 are equal and opposite. The curve of the difference, IV, 
 however, approaches a circular orbit of double the di- 
 ameter of the final orbit of the electrons, which is shown 
 by the larger circle. 
 
 It is to be regretted that these curves have not been 
 computed for a greater distance than they have been. 
 This may, of course, be done, but the series which have 
 been developed will have to be abandoned on account of 
 their non-convergence. By omitting the first periodic 
 term from the series, which has the greatest frequency, and 
 by computing it separately, and then computing the rest 
 of the series as we have done above, the result may be 
 considerably extended in time. The computation of the 
 first term may be added to the rest after they are sep- 
 
The Atom 109 
 
 arately obtained. The labor involved in these computa- 
 tions is, however, considerable, as will be evident from 
 an inspection of the work involved in the one example 
 that has been given. 
 
 It will be interesting to observe the differences be- 
 tween the curves obtained from the different spectral 
 series. The example that has been computed is the case 
 of the head series only. It is necessary to content our- 
 selves with this one example at present. The next case 
 that would naturally be computed is that where T2 = i. 
 
XIII 
 
 |T is proposed in the following sections to give 
 some account of the results which have been 
 obtained by considering atoms in their first 
 state when neither radiating nor absorbing 
 energy. It will be recognized at once that, if we knew 
 the correct expression for the mechanical force with 
 which one moving electrical charge acts upon another 
 for any kind of motion, it should be possible to assume 
 that the motion is circular motion such as we suppose the 
 electrons have in the normal undisturbed state of all 
 atoms. By applying these results to the electrons in 
 the atoms it is conceivable that we may by adding up 
 the effects of the individual electrons in an atom event- 
 ually arrive at the nature of the forces that atoms exert 
 upon each other. The fundamental problem is, therefore, 
 to obtain an expression for the mechanical force that 
 a single electron revolving in a circle exerts upon another 
 revolving in a different circle of different radius and dif- 
 ferent frequency. 
 
 The author has solved this problem by the use of two 
 of the forms of electromagnetic theory that have been 
 proposed, first * by the equations given by J. J. Thomson 
 in his paper of 1881, and second^ by the equations due 
 
 •A. C. Crehore, Phil. Mag., Vol. XXVI, July, 1913, p. 58. Also 
 Phil. Mag.. Vol. XXIX, June, 19 15, p. 750; Phil. Mag., Vol. XXX, 
 August, 1915, p. 257. 
 
 'A. C. Crehore, Pbys. Rev., N. S., Vol. IX, No. 6, June, 1917, 
 p. 445. 
 
 no 
 
The Atom in 
 
 to Lorentz, which represent the current form of the 
 equations of this theory. Electromagnetic theory has 
 passed through several important stages of develop- 
 ment since the early days when Maxwell pubhshed his 
 celebrated treatise. And the process of this develop- 
 ment is not at an end by any means as yet. It should 
 not end until the results obtained from the theory are 
 in complete harmony with all the facts of observation. 
 A recent valuable contribution' has been made to this 
 theory by Mega Nad Saha, who makes use of the modern 
 four-dimensional analysis of Minkowski. This investi- 
 gator arrives at equations having greater generality than 
 those of Lorentz, which seem hkely to have an important 
 bearing upon the problem before us. We shall outline 
 the results obtained by the author by the use of the 
 Lorentz form of equations only, omitting any reference 
 to the Thomson equations. The possible modifications 
 that will be permitted by the use of the Saha equations 
 have not yet been investigated. Reference must be had 
 to the original publications for a detailed account of the 
 derivation of the equations which we shall use here merely 
 as the result there obtained. The complete equation'' 
 expressing the force that a second electron revolving in 
 a circular orbit exerts upon a first electron in another 
 orbit is too long to repeat here. The force is a variable 
 force with time as the two electrons revolve about their 
 orbits. If, however, these two electrons are in fixed 
 orbits in atoms, the eff"ect which they individually con- 
 tribute to the attraction or the repulsion of the atoms 
 must depend upon the average value of the variable 
 force, averaged for time. The average force obtained 
 from this equation, when resolved along the center line 
 
 ' Mega Nad Saha, loc. cit. 
 
 ^ Loc. cit., equations (48), (49), and (50), pp. 453, 454. 
 
112 The Atom 
 
 of the two orbits, is a very simple expression, namely : '■ 
 
 Fr = le^Mi - (- ^ sin a + Z cos ay^r''' . (196) 
 
 This denotes the force that the second electron exerts 
 upon the first. The ^2 is the velocity of the second 
 electron and e is the charge, r is the distance between 
 the centers of the orbits supposed to be fixed and constant. 
 The angle a is the angle between the directions of their 
 axes of revolution, and the X and Z are the direction 
 cosines defining the position of the center of the orbit 
 of the second electron with reference to a set of rectangular 
 axes, i, j, and k, through the center of the orbit of the 
 first electron. 
 
 It is to be remarked first that the velocity of the 
 first electron does not appear in this equation at all. 
 That is to say, the force upon the first electron is, ac- 
 cording to this result, entirely independent of its own 
 velocity, and it does not matter what it is doing. As a 
 direct result of this it may be shown that the force upon 
 the first electron due to the second one may be entirely 
 different from the force exerted upon the second due to 
 the first. We would obtain the force on the second elec- 
 tron due to the first by putting jSi in place of jSj in the 
 equation, and, if they were not equal to each other, the 
 action and reaction would be unequal. It should be 
 stated here that this equation gives the forces due to the 
 velocity of motion of the electrons only, and becomes 
 zero when the velocity is zero. There are besides these 
 forces the large electrostatic forces * which have purposely 
 been omitted from the equation because the electrostatic 
 part cancels out when the positive nucleus of the atoms 
 is also taken into the account. 
 
 The equation represents only the first term of an in- 
 
 ^ Loc. cit., equation (54), p. 456. ' Loc, cit. See top of p. 456. 
 
The Atom 113 
 
 finite series of terms, r~', r~*, etc., and at great distances 
 the third and higher powers of r become so small that all 
 terms except the first are negligible, that is to say, the 
 equation as it stands is supposed to apply to the two 
 electrons only when they are at a great distance apart 
 as compared with the diameters of their orbits. There 
 is one more quahfication that has to be made as to this 
 equation. The average given in (196) was obtained 
 under the supposition ^ that the Doppler factor, 
 
 ^= a^ = '-^' • • ■ • (^97) 
 
 is a constant so nearly equal to unity that it cannot 
 affect the average, for i/A* occurs as a factor of the 
 original equation. On this point the author has been 
 taken to task in a long article by G. A. Schott,^ who has 
 shown that the supposition that the Doppler factor is 
 unity gives a different average force ^ from the supposition 
 that it is variable, as given in the equation last above. 
 On the hypothesis, which the author made, that this 
 factor is sensibly equal to unity, Schott has also verified 
 the result given above in (196). The result obtained by 
 Schott on the supposition of A variable also gives a force 
 that varies as the inverse square of the distance, but the 
 magnitude differs in sign and depends upon j8* instead 
 of j8^ as in the equation given above. 
 
 The chief result, which has now been established by 
 means of these deductions from the current form of 
 electromagnetic theory by these investigations, is that 
 
 ' hoc. aX. See top of p. 455. 
 
 2 G. A. Schott, Phys. Rev., Sec. Ser., Vol. XII, July, 1918, p. 23. 
 See also the author's reply to Schott, Pbys. Rev., Vol. XIII, No. 2, 
 February, 19 19, p. 89. 
 
 ' Loc. cit., the Schott paper, p. 37, equation (50). Also p. 91, 
 equation (i), author's reply. 
 
114 ^^^ Atom 
 
 this theory demands that there be a force, which one 
 revolving electron exerts upon another at a great distance, 
 omitting all electrostatic forces because they eventually 
 cancel, that shall vary as the inverse square of the dis- 
 tance law. If this result is applied to a great multitude 
 of electrons such as make up the sum total of all the 
 electrons in all of the atoms of a material mass of matter, 
 the process of summing up these forces does not change 
 in any way the character of the law of variation of the 
 force with the distance. It may, therefore, be said that 
 the above investigation demands that there be a force 
 between two material massive bodies at a great distance 
 from each other that varies inversely as the square of the 
 distance. Now, the chief force that we know actually 
 exists between any two bodies at a distance is the gravi- 
 tational force which obeys the inverse square of the 
 distance law. If, therefore, these deductions from the 
 theory are not in harmony in all respects with the gravi- 
 tational force, we are forced to conclude that something 
 is amiss with the theory. On this account the author 
 has made a careful comparison between the result of the 
 theory and the actual gravitational force which is known 
 with precision. It is very significant indeed that the 
 equation (196) above is in complete agreement with the 
 gravitational law* in every respect but one, and this is 
 the magnitude of the force. There are several other 
 checks besides the magnitude of the force that it must 
 fulfil. They are as follows. It must make the force 
 proportional to the product of the masses; it must show 
 that the force is always an attraction and never a re- 
 pulsion; it must show that the force is independent of 
 the orientation of the two bodies, whether they be crystals 
 
 » A. C. Crehore, Pbys. Rev., N. S., Vol. XII, No. i, July, 1918, 
 p. 13. 
 
The Atom 115 
 
 or any other of the forms of matter, solids, liquids, or 
 gases. 
 
 It will presently be shown that this equation agrees 
 in all these respects in a remarkable manner with the law 
 of gravitation, except only in the magnitude; and the 
 Schott equation, obtained by assuming the Doppler 
 factor variable, does not agree in any respect save that 
 it requires the inverse square of the distance law. 
 
 As to the magnitude of the force the equation is evi- 
 dently deficient because it does not agree with one of 
 the most fundamental laws of Nature, one of Newton's 
 laws, that of equal Action and Reaction. The equation 
 as it stands makes it possible that the attraction of the 
 body A for the body B may differ from the attraction of 
 the body B for the body A. This is because the two 
 velocities j8i and 182 do not occur symmetrically in the 
 equation. If it is found that we can make use of this equa- 
 tion, and, by the simple expedient of correcting it by the 
 use of a constant multiplier, make it agree with gravita- 
 tional law in all respects, then there are strong grounds 
 for believing that this constant multipher should have 
 existed in the correct form of electromagnetic equations 
 from which this was derived. And in this manner it is 
 hoped that this experimental check of the electromagnetic 
 theory by comparing its results with known facts may be 
 of material assistance in eventually revising the theory. 
 
 We shall, therefore, arbitrarily introduce a multiply- 
 ing factor to correct the magnitude of the force expressed 
 in (196) and shall then proceed to determine the required 
 numerical value of this factor to make it agree with the 
 gravitational law. However, it is conjectured that one 
 of the factors must be j8i^ in order to make the equation 
 conform to the law of equal Action and Reaction. Let 
 us temporarily denote the rest of the multiolying factor 
 
ii6 The Atom 
 
 by X, signifying an unknown quantity to be determined, 
 and make the whole factor ^i'x. To anticipate the result 
 of the determination of x, it may now be stated that 
 numerically x comes out equal to .8625 x io~^'. This 
 value is so close to the value of the mass of the electron 
 itself given in (167) above, namely .898 X lO"'" that 
 we have strong grounds for thinking that the true mul- 
 tiplying factor should be mo/3i^. 
 
 If there should be any multiplying factor at all re- 
 quired for this equation, we must expect that it will have 
 some value connecting it in a very simple manner with the 
 properties of the electron, and this factor satisfies this 
 demand in a very complete manner. But, if this is the 
 true factor, it immediately raises the question, how is it 
 possible to multiply the expression on the right of equa- 
 tion (196) by the quantity moPi^ and still have the ex- 
 pression represent a force. For the quantities on the right 
 of the equation should already have the dimensions of a 
 force, and the result of the multiplication is to make the 
 dimensions a force times a mass. This is a most important 
 consideration, indeed, and it leads again to some most 
 important results, as will be shown. 
 
 It was shown above, in considering the Lorentz mass 
 formula (144), that it is customary by writers on the 
 modern electromagnetic theory to suppress the quantity 
 k, the specific inductive capacity of the medium, regard- 
 ing it as equal to unity and as being dimensionless. We 
 corrected this equation by introducing the k, writing it 
 as in (169) in order to make the dimensions of the two 
 members of this equation the same, on the supposition 
 that k is not dimensionless, but that it has dimensions 
 in terms of length and time. And this necessitated that 
 ak on the left of the equation (161) should have the dimen- 
 sions of the reciprocal of the Rydberg constant, namely 
 
The Atom 117 
 
 that of time alone. This result, making the radius 
 times the specific inductive capacity have the dimensions 
 of time, required that the dimensions of k should be those 
 of the reciprocal of a velocity. This is a rational result 
 because we already know that the product of k and fj,, 
 the magnetic permeability, are those of the reciprocal 
 of the square of a velocity, namely the velocity of light, 
 for the equation 
 
 feM=^, (198) 
 
 has been known for some time; but neither the dimensions 
 of k nor of fj. separately have been known. The fact 
 that we possess this equation for the product of k and /x 
 in itself shows that both k and (j, should have some dimen- 
 sions in terms of length and of time. And since we have 
 fixed upon the dimensions of k as being those of the 
 reciprocal of a velocity, the dimensions of fx in terms of 
 length and of time are automatically determined by the 
 equation (198). This gives jjl the same dimensions as 
 k, namely the reciprocal of a velocity. It also makes the 
 ratio of fe to /i dimensionless, since it comes out the ratio of 
 two velocities, thus having the same kind of dimensions as 
 /S, the ratio of the velocity of an electron to that of light. 
 
 This matter has been referred to again here because an 
 examination of the original electromagnetic equation, 
 from which (196) has been derived, shows that the k 
 has again been suppressed. This becomes apparent when 
 we write out the dimensions of the quantities on the 
 right of the equation, which are the same evidently as 
 the dimensions of e^/r". 
 
 The dimensions of e on the electrostatic system of 
 units are given by (156) above, and the dimensions of 
 e*/r^ are. therefore, 
 
 eVr^ = LMT-% (199) 
 
ii8 The Atom 
 
 The dimensions of force are 
 
 F = LMT-^ (200) 
 
 The two members of the equation as it now stands do 
 not, therefore, have the same dimensions. Some mul- 
 tiplying factor having some physical dimensions is in 
 fact required in order to correct the dimensions of the 
 equation, assuming that k is not dimensionless in terms 
 of L and T. This factor we shall assume is the x^i\ 
 or its equivalent, as we shall prove, mo/Si^ having the 
 the dimensions of mass. The complete revised equation 
 now becomes 
 
 Fr = ^e^xfii'Piii -i-Xsina + Z cos ay2r-^ (201) 
 
 where the x may be taken as equivalent to the mass of 
 the electron, mo. 
 
 The dimensions of the quantities on the right of the 
 equation are the same as the dimensions of ehno/r\ and 
 these dimensions according to (199) are 
 
 LMT-mk (202) 
 
 But these dimensions must be those of force, LMT-^. 
 Hence Mk must have zero dimensions ' in terms of L and 
 T. But we have already made the dimensions of k 
 those of the reciprocal of a velocity. Hence, to satisfy 
 this equation, the dimensions of mass must be the re- 
 ciprocal of k and equal to those of velocity, namely 
 LT-\ 
 
 We have by these means thus come to the conclusion 
 that mass, specific inductive capacity, and magnetic 
 permeability are not fundamental units like length and 
 time, but that they may be expressed in terms of length 
 and time. Very strong confirmation of these ideas is to 
 
 ^ Loc. cit., Pbys. Rev., June, 1917, p. 464, equation (77). 
 
The Atom 119 
 
 be found in the results obtained by eliminating M, k 
 and n from the common tables of dimensions of quantities 
 as ordinarily given in an electrostatic system and in an 
 electromagnetic system. Such a table is given on the 
 following page for some of the more common units, and 
 a reduction is made to a new system which may be called 
 the L-T, or the space-time, system, in which all dimen- 
 sions of every kind of quantity are expressed merely in 
 terms of space and time. 
 
 The reduction of a unit in either system, electro- 
 static or electromagnetic, to the space-time system is 
 effected by substituting the values above determined, 
 
 namely, 
 
 k = fi = Lr^T (203) 
 
 and M = LT-i (204) 
 
 It is to be noticed, first, that each unit, whether re- 
 duced from the electrostatic system or from the electro- 
 magnetic system comes out of the same dimensions in 
 terms of L and T. This is as it should be if we are to 
 r^ard dimensions as characteristic of a quantity. In- 
 deed, the common systems are misleading in regard to 
 this and make it appear that there is nothing definite 
 about the dimensions so far as length and time are con- 
 cerned, although this is in appearance only. 
 
 In the second place it will be noticed that several of 
 the quantities which have formerly been regarded as dif- 
 ferent things have the same dimensions in the space- 
 time system. And these quantities so reduced to the 
 same dimensions do not come out in a haphazard fashion, 
 but they are the very quantities that we have already 
 suspected were of the same nature and therefore might 
 be expected to have the same dimensions. For example, 
 quantity of electricity, or electrical charge, and quantity 
 of magnetism, or the strength of a magnetic pole, receive 
 
120 
 
 The Atom 
 
 
 
 Dimensions of 
 
 Dimensions of 
 
 
 Dimensions of 
 
 
 ■a 
 
 electrostatic 
 
 electromagnetic 
 
 Ratio electrostatic to 
 
 length and time 
 
 Kind of 
 
 S 
 
 system of units. 
 
 system of units. 
 
 electromagnetic units. 
 
 system of units. 
 
 Quantity 
 
 1 
 
 Exponents only 
 
 Exponents only 
 
 Exponents only 
 
 Exponents 
 
 
 expressed. 
 
 expressed. 
 
 expressed. 
 
 expressed. 
 
 
 m 
 
 LM T h 
 
 LM T 11 
 
 LM T k ti 
 
 L T 
 
 Mass 
 
 O I o o 
 
 01 
 
 00 00 
 
 1 — I 
 
 Specific 
 
 
 
 
 
 
 inductive 
 
 
 
 
 
 
 capacity. 
 
 K 
 
 CO O I 
 
 — 2 2—1 
 
 20—2 I I 
 
 — I I 
 
 Permea- 
 
 
 
 
 
 
 biUty... 
 
 M 
 
 —20 2—1 
 
 00 I 
 
 — 20 2— I— I 
 
 — I 1 
 
 Momentum 
 
 mv 
 
 I I — I 
 
 I I — 1 
 
 00 00 
 
 2 _ 2 
 
 Moment of 
 
 
 
 
 
 
 momentum 
 
 mva 
 
 2 I — I 
 
 2 I — I 
 
 00 
 
 3 — 2 
 
 Force 
 
 
 I I — 2 
 
 11—2 
 
 00 
 
 3 — 3 
 
 Energy 
 
 Electric 
 
 capacity 
 
 mv^ 
 
 2 1—2 
 
 21—2 
 
 00 00 
 
 3 -3, 
 
 C 
 
 10 I 
 
 —10 2—1 
 
 20-2 I I 
 
 X 
 
 
 Magnetic 
 
 
 
 
 
 
 
 self-in- 
 
 
 
 
 
 
 
 duction . 
 
 L 
 
 -10 S — I 
 
 10 
 
 -20 2—1—1 
 
 I 
 
 
 Electric 
 
 
 
 
 
 
 resistance 
 
 R 
 
 — 10 I — I 
 
 I — I 1 
 
 — 20 2— I— I 
 
 
 
 Electro- 
 
 
 
 
 
 
 motive 
 
 
 
 
 
 
 force 
 
 E 
 
 H -I -J 
 
 Si -2 i 
 
 -10 I -i -J 
 
 } — 2 
 
 
 Electric 
 
 
 
 
 
 
 
 current. . 
 
 I 
 
 IJ -2 J 
 
 ii -I -J 
 
 1 -I i 4 
 
 1 — 2 
 
 
 Magneto- 
 
 
 
 
 
 
 
 motive 
 
 
 
 
 
 
 
 force. . . , 
 
 
 3i -2 i 
 
 JJ -I -i 
 
 1 -I i J 
 
 i — 2 
 
 
 Electric 
 
 
 
 
 
 
 force 
 
 
 -ii -I -i 
 
 H -2 J 
 
 -10 I -} -i 
 
 4 — 2 
 
 Magnetic 
 
 
 
 
 
 
 force 
 
 H 
 
 ii-2 J 
 
 -JJ -I -i 
 
 1 -I i J 
 
 4 — 2 
 
 Electric 
 
 
 
 
 
 
 displace- 
 
 
 
 
 
 
 ment 
 
 D 
 
 -Ji-I i 
 
 -ii o-J 
 
 1 -I i J 
 
 — J — I 
 
 
 Magnetic 
 
 
 
 
 
 
 
 flux den- 
 
 
 
 
 
 
 
 sity or in- 
 
 
 
 
 
 
 
 duction . 
 
 B 
 
 -li o-i 
 
 -H-i J 
 
 -10 I -J -} 
 
 — } — I 
 
 
 Electric 
 
 
 
 
 
 
 quantity 
 
 Q 
 
 5i -I i 
 
 ii -i 
 
 10 -I i J 
 
 i — X 
 
 
 Total mag- 
 
 
 
 
 
 
 
 netic flux 
 
 Z 
 
 Ji o-J 
 
 H -I i 
 
 -10 I -J -i 
 
 1 — I 
 
 
 Quantity 
 
 
 
 
 
 
 
 magnet- 
 
 
 
 
 
 
 
 ism . ... 
 
 
 U o-i 
 
 -I i 
 
 — 10 I — } — J 
 
 i — I, 
 
 Planck's 
 
 
 
 constant 
 
 h 
 
 2 1 — I 
 
 2 I — I 
 
 00 
 
 3 — 2 
 
 Rydberg's 
 
 
 
 
 
 
 constant 
 
 K 
 
 00—1 
 
 DO — I 
 
 00 
 
 — 1 
 
 Newtonian 
 
 
 
 
 
 
 constant 
 
 k 
 
 3—1—2 
 
 3 — 1—2 
 
 00 00 
 
 2 — I 
 
 the same dimensions in the space-time system, indicating 
 that they are of the same nature. And, again, electric 
 capacity and the coefficient of self- or mutual induction 
 
The Atom 121 
 
 receive the same dimensions, that of a time. Elec- 
 tromotive force receives the same dimensions as 
 magnetomotive force. Electric resistance comes out 
 dimensionless, being the ratio between two velocities 
 like the quantity /S, which is the ratio of the velocity of an 
 electron to the velocity of light. This fact gives electric 
 current the same dimensions as electromotive force because 
 of the relation in Ohm's law, R = E/I. And, again, elec- 
 tric force and magnetic force have the same dimensions. 
 So, also, do electric flux, or displacement, and magnetic 
 flux density come out of the same dimensions. Energy 
 comes out as the cube of a velocity, which becomes 
 rational when we regard energy as equivalent to mi^ 
 and mass as a velocity. 
 
 It may at present be difficult to obtain any mental 
 picture from -these dimensional formulae for the diff'erent 
 kinds of quantities, but it is maintained that it was stiH 
 more difficult when the dimensions of these quantities 
 were expressed in two independent systems, the electro- 
 static and the electromagnetic. Without having some 
 mental picture of the quantities k and fi, it was im- 
 possible to harmonize the very different-looking dimen- 
 sions of the very same quantity as expressed in the 
 two systems. This table of dimensions makes an appeal 
 to reason in such a strong way that it is regarded as 
 strong support for the ideas that have led to the deter- 
 mination of the dimensions of mass, specific inductive 
 capacity, and magnetic permeability, that is to say, in 
 support of the theories here advanced. 
 
XIV 
 
 |ET us next return to the equation derived from 
 electromagnetic theory but modified by the 
 factor Xj8i^ or mojSi^ in (201). The presence 
 of this factor makes a tremendous difference 
 in the magnitude of the force, since mo itself is about 
 .9 X 10"^'' and the /3i* of the order of io~*. Let us in 
 advance of the matters which follow admit that there is 
 very great probability, on account of the results obtained, 
 that the original form of electromagnetic theory will 
 have to be changed in some way to fit the case. Just 
 what the change in the original equations will be is dif- 
 ficult to say on this evidence alone, but an attempt ^ has 
 been made in the original article in which this equation 
 was derived to trace back to their origin the particular 
 terms in the fundamental expression of electromagnetic 
 theory that give rise to the force varying as the inverse 
 square of the distance. For there are many other terms 
 in the original equation which have no effect at all so far 
 as the inverse square of the distance terms are concerned. 
 It was there pointed out that all of the inverse square 
 of the distance terms in this so-called gravitational 
 equation arise either from the differentiation of the 
 scalar or vector potential, </> or a, with respect to the time 
 in distinction to the space coSrdinates. 
 
 This matter seems to be very significant now in the 
 light of the recent papers by Saha above referred to. 
 For he has shown that in the so-called Doppler factor, 
 
 ' Loc. cit., Pbys. Rev., June, 1917, p. 464. 
 122 
 
The Atom 123 
 
 (197) above, the dr should be replaced by a generalized 
 value depending equally upon the four coordinates in 
 the generalized Minkowski space, which includes x, y, 
 and z as well as t. This change will of necessity make a 
 difference in the value of the Doppler factor. At the 
 present time it is not possible to say what it will become, 
 but the point is that there exist to-day good grounds 
 for supposing that it is different from the expression al- 
 ready given in (197), in which the differentiation is with 
 respect to the time only. If we suppose that a similar 
 factor having a small value, such as mojSi^ should be in- 
 troduced into the second term of the Doppler factor, 
 then we are entirely justified in considering that the factor 
 is sensibly equal to unity, as it was assumed to be in the 
 derivation of the gravitational equation (196). Also 
 the work of Schott above referred to, which does not 
 lead to a correct gravitational form, cannot be a correct 
 result on the changed premises, namely that the Doppler 
 factor has undergone a change. It should be an argu- 
 ment in favor of a revision of this Doppler factor that 
 the results obtained regarding it as sensibly equal to 
 unity agree in every respect with the gravitational law, 
 after we have introduced the factor, mojSi^ But legiti- 
 mate grounds have appeared through the work of Saha 
 to modify the form of the Doppler factor independently 
 of any other considerations. 
 
 Let us now apply the equation to find the attraction 
 between two material bodies at a great distance apart. 
 As it stands, the equation represents the force between 
 a single pair of electrons, one in each of the two bodies. 
 The only quantities that have any different values in 
 the equation when applied to a second pair of electrons 
 are the velocities, j8i and 182, the angle, a, and direction 
 cosines, X and Z, which occur within the bracket. The 
 
124 The Atom 
 
 distance, r, may be regarded as so great in comparison 
 with the size of the body that r is very approximately 
 the same for any two pairs of electrons in the bodies, 
 taking one from each body. 
 
 It is evident that the force will be different between 
 the same pair of electrons having the same radius of orbit 
 and same angular velocity, depending upon how the 
 planes of the orbits are turned with respect to each other. 
 In any body save a crystal the chances are that there 
 will be orbits turned in every possible manner with re- 
 spect to each other, and it therefore becomes desirable 
 to find the average force between a single pair of electrons 
 as they are turned in every conceivable manner without 
 changing the positions of the centers of the orbits. We 
 have already taken a time average of the force as the 
 electrons proceed around their orbits, and now we want 
 a space average of the force as the orbits themselves are 
 turned in all possible ways. The process of obtaining this 
 space average is given in Appendix A so as not to divert 
 attention from the main argument at present, and the 
 result of it is that the space average of the force is ob- 
 tained by simply replacing the bracket in equation (201) 
 by the numeric f , giving the result as follows, 
 
 Fr = h'x^i%''r-^ (205) 
 
 If, therefore, we write down the force of attraction 
 between just one electron in the body, i, and each of 
 the electrons in the second body, 2, and add them all 
 together, we would have an equation just like (205) in 
 
 which the 0S is replaced by SjS^ taken over every electron 
 
 2 
 
 in body 2. And, again, if we write down the attraction 
 of the whole body, 2, for each electron in the body, i, 
 the total is expressed by an equation like (205) in which 
 jSi^ is replaced by S/3^ the summations being extended 
 
The Atom 125 
 
 over all of the electrons in each of the bodies. This 
 equation is as follows, 
 
 F =1 e=xSj82S/3V-2 (206) 
 
 1 2 
 
 No further progress can be made toward getting a 
 numerical value of the attraction without possessing 
 some knowledge of the velocities of the electrons in the 
 rings of electrons in the various atoms that enter into 
 the material body. Use may now be made of the formula 
 developed above for the velocity of the electrons in any 
 ring of electrons in (172). Squaring this, we obtain j8^ 
 for a single electron in the ring, and multiplying by the 
 number of electrons in the ring, p, we find the sum of 
 the squares of /8 for a ring of electrons, namely 
 
 To simplify matters as much as possible, let us con- 
 sider the attraction between two atoms of the simplest 
 kind, each atom having but a single ring of electrons as 
 in hydrogen or hehum. If the two bodies, i and 2, are 
 alike, each being an atom of hydrogen, say, then the 
 total value of the 2/8^ for each body is just the expression 
 
 in (207), and the product S;8^S/3^ is simply the square of 
 
 I 2 
 
 (207). Substituting this in (206) gives the complete 
 
 force between the two atoms of hydrogen as 
 
 F = l^K^^r-^ (208) 
 
 16 ehni' 
 
 This should represent the attraction on the average 
 between the two hydrogen atoms or two helium atoms 
 when the correct numbers of electrons in the rings are 
 substituted for p in the two cases respectively. 
 
126 The Atom 
 
 But the attraction on the average between two atoms 
 of hydrogen is given by Newton's gravitational law to 
 be 
 
 F = kmsh-^ (209) 
 
 By equating the two expressions for the same force, 
 (208) and (209), we may find a value for the gravitational 
 constant, k, and thus see that the two expressions are 
 really equivalent. We have 
 
 3 16 Vemo/ 
 
 which gives a value for the Newtonian constant in terms 
 of the properties of the electrons with the exception of 
 the unknown quantities x and p, the latter being the 
 number of electrons in the hydrogen atom. We shall 
 take this number of electrons in the hydrogen atom in 
 its normal condition as equal to 2 and solve the equation 
 for X, giving 
 
 '"^■'' (211) 
 
 'HtJ 
 
 Using the current values of e, mo and h, namely 
 
 e = 4.774 X io~'" 
 mo = .90 X 10"^' 
 b = 6.54.7 X io~^^ 
 
 and taking the Newtonian constant as equal to 666 x lo"^", 
 the value oix becomes 
 
 X = .8625 X 10"^' (212) 
 
 This numerical value is so close to the value of the 
 mass of the electron in view of the uncertainties in the 
 experimental values used in determining it that there are 
 good grounds for thinking that the multiplier, x, should 
 have been the very simple physical quantity, mo, the mass 
 
The Atom 127 
 
 of the electron. It should be pointed out that there is 
 some uncertainty in the value of the Newtonian constant. 
 Astronomers usually take the mass of the earth as unity, 
 but to obtain the constant on the C.G.S. system of units 
 really involves the mass of the earth in grams, and 
 it is difficult to say with assurance how much error 
 there is in the above numerical value of k. The values 
 of e and of b used above are those determined by Millikan, 
 Substituting mo for x in (211) we obtain the expression 
 for the gravitational constant 
 
 ^=i£^o ^"'3) 
 
 The dimensions of this expression are correct. The 
 dimensions of h are UMT-^, and of b'^/ehtio in the electro- 
 static system, Lhr^, and in the space-time system, L?T^^, 
 which is equivalent to those of the Newtonian constant. 
 
 The Newtonian equation of gravitation is 
 
 F = kmimir~^, 
 whence k = F(rnimi)-h^. 
 
 The dimensions of force are LMT^, and, according 
 to this value of k, the dimensions of it ar6 L^M~^T-K 
 Replacing Af-i by its equivalent, L~^T, we have the 
 dimensions of k, L?T-\ which agrees with (213) above. 
 Or we might have replaced the M"' by its equivalent, k, 
 the specific inductive capacity, and obtained UJ-^k, 
 which becomes equivalent to the dimensions on the 
 electrostatic system obtained above, namely Lfe-^ re- 
 membering that DT-^ is equivalent to fe-". Although 
 we have employed the same symbol, k, for both the 
 Newtonian constant and for the specific inductive capacity 
 in this work it is difficult to see how any confusion can 
 arise from this. 
 
128 The Atom 
 
 The expression obtained in an article previously pub- 
 lished^ for the Newtonian constant was a much more 
 complicated one than that given above, namely 
 
 This was obtained through the use of the Bohr value 
 of the Rydberg constant, which gives a numerical value 
 in close agreement with the above. The quantities tt*, 
 c* and niH^ occur in this in addition to the quantities that 
 occur in the simpler expression (213). The specific 
 inductive capacity is denoted by k' here, since the two 
 k's occur in this equation. The powers of the quantities 
 are very high, that of e being the tenth power. The 
 simplicity of the value in (213) is greatly in its favor. 
 Moreover, the dimensions of this expression do not agree 
 with those of the Newtonian constant, even if we take 
 the specific inductive capacity as the reciprocal of a mass. 
 This fact strengthens the grounds for the opinion that 
 the Bohr expression for the Rydberg constant does not 
 represent a true equation between physical quantities 
 when the k is omitted. 
 
 Denoting by mi and 7712 the masses of two bodies, their 
 attraction for each other is by Newton's law, 
 
 F = kmimir-^ (215) 
 
 Substituting the value of k in (213) above, gives 
 
 ^=ii4'"^"^''~' ^"^'^ 
 
 And by (206) we have 
 
 F = le'^oSiS^SiSV-^ (217) 
 
 I 2 
 
 ^ Loc. cit., Pbys, Rev., July, 1918, p. 15, equation (9), 
 
The Atom 129 
 
 Equating (216) and (217) gives 
 
 m,m = ^'S/3^S|3^ (2i8) 
 
 from which we derive the value of mi or ma, namely 
 
 ^ = £!pS)8^ (219) 
 
 The dimensions of this expression for the mass of a 
 body in general are correct, for 2/8^ has no dimensions, 
 and on the electrostatic system e^/h has the dimensions 
 LT~^k, and putting the k equal to Lr^T, the dimensions 
 of e^/b become zero on the space-time system, thus re- 
 ducing the expression on the right of (219) to the dimen- 
 sions of mass. This tells us that the mass of a body is 
 proportional to the sum of the squares of the velocities 
 of all the electrons in the body, that is, proportional to 
 the kinetic energy of all the electrons within the body. 
 
 The expression for the mass of a body as given in (219) 
 may seem surprising at first, since we attribute the mass 
 of a body to the sum of the masses of the nuclei of all 
 the atoms in the body, and the nucleus makes no appear- 
 ance in this expression. It should be remembered that 
 the masses of bodies are strictly proportional to their 
 weights, and the expression in (219) is derived from the 
 weight of the body primarily. There is great probability 
 that the expression for the mass derived from summing 
 up the masses of the separate nuclei of the atoms in a 
 body will come out equivalent to the expression in (219) 
 because the sum of the electrical charges is the same for 
 the nuclei as for the total number of electrons. The 
 Lorentz mass formula for one atomic nucleus is 
 
 4 g 
 "* ~ 5 c^ak' 
 
130 The Atom 
 
 where E denotes the positive charge of the nucleus. If 
 the number of atoms in the body in question is A, many 
 of the atoms being of different kinds so that the radii, 
 a, differ, the total mass of the body is 
 
 A E^ 
 A ^c^Ra a 
 
 There is no reason to suppose that any different value 
 would be obtained from this summation than that given 
 above in (219), although the summation may be difficult 
 to obtain except in special cases. If the formula is applied 
 to the simple case of hydrogen, where the atoms are all 
 alike, and where E = 2e, and where the radius, a, is the 
 same in every atom, namely, according to (152) and (148) 
 above 
 
 '^=i = 3.2^^ = 4.8620x10-. .... (153) 
 we have the mass of the hydrogen gas 
 
 A Sc^Ka a s^k 3.2 
 
 which is evidently equal to the mass of the gas because it 
 is the mass of one atom times the number of atoms. 
 
 The formula (219) gives an equal result, for by (207) 
 the sum of ^ for one atom is hniH/e^mo, and for A atoms 
 is A times this. When this value is substituted in (219), 
 the coefficient, ehno/h cancels, leaving only iuhA for the 
 mass of the gas, the same expression as obtained from 
 the nucleus of the atom. 
 
 It is of interest to note that we have just pointed out 
 that e^ and h have the same dimensions in the space- 
 time system, and that each is the same as a moment of 
 momentum, L^T~^. From this it seems likely that h 
 is closely connected with e and is constant because e is 
 
The Atom 131 
 
 constant. And again it may be conjectured that the 
 energy content of the negative electron and the positive 
 hydrogen nucleus are as follows, for we may write the 
 Lorentz mass equation 
 
 rriHC^ = ^-A- = 1.658 X iO-2< X 9 X lO^" = 1.492 X IQ-', 
 SanK 
 
 moc^ = -^ = .898 X 10-" X 9 X io2» = 8.082 X 10-'. 
 
 The dimensions of each of the members of these equa- 
 tions are those of energy, or the cube of a velocity, L^T~\ 
 and the expression e''/ak on the right of the equations is 
 of the form of potential energy. We may think of these 
 figures as representing the energy contained in the hydro- 
 gen nucleus, 1.492 x io~' ergs, and the electron itself, 
 8.082 X io~^ ergs, the former being about 1845 times 
 the latter. As compared with the energy, bK, which 
 represents the order of magnitude of kinetic energy of 
 the electrons in their orbits, namely .2154 x lo"^" ergs, 
 the energy content of the electron itself is very large, 
 37,500 times greater. 
 
XV 
 
 F the gravitational equation (217) is applied 
 to the earth as one of the bodies and to a 
 second body on the surface of the earth, this 
 equation then expresses the weight of that 
 body. When different bodies are substituted for the 
 one, the earth remaining as the common body, it is 
 evident that the weight will vary as the sum of the squares 
 of the velocities of all the electrons in the body, and 
 hence the weight is proportional to the mass as it is known 
 to be in fact. 
 
 The equation enables us to write down the weight of 
 any kind of a body small or large, and it shows that 
 the weight contributed to any single atom by one of the 
 rings of electrons in it depends merely upon the sum of 
 the squares of the velocities of the electrons in the ring. 
 We have made the velocity of the electrons in any ring 
 depend only upon the number of electrons in the ring in 
 equation (172) above, no matter how many other rings 
 of electrons there may happen to be in the same atom, 
 but this is regarded only as a first approximation, the 
 velocity also depending upon the radius of the orbit so 
 far as second order terms are concerned. Let us then 
 apply the gravitational equation to write down the 
 weights of rings of electrons only. Now one body is 
 the earth and the other body is a single ring of electrons 
 on its surface. The average weight of the ring of p 
 electrons when oriented in all possible ways is then 
 
 . . . (220) 
 
 p E 
 132 
 
The Atom 
 
 133 
 
 where Sj3^ refers to the ring of p electrons only and S/8^ 
 
 p E 
 
 refers to all the electrons composing the mass of the earth, 
 r£ becoming the radius of the earth. If, therefore, we 
 compare the weights of two rings of electrons having 
 diflFerent numbers in the rings, and use the expression 
 
 x^a, _p'bmn 
 p 4 e'mo 
 
 given in (207) above for the ring, it is apparent that the 
 only variable quantity that changes when one ring is sub- 
 stituted for another is the p\ for the other quantities in 
 this expression are constants, and, of course, the unknown 
 quantities pertaining to the earth, the other body, do 
 not change. It may be stated, therefore, that the weights 
 of rings of electrons are proportional to the squares of 
 the numbers of electrons they contain. The absolute 
 weights may then be written down as soon as the weight 
 of some one ring is known. Let us say that the weight 
 of a ring of two electrons is the same as the weight of a 
 hydrogen atom, and take it as 1.008 on the scale, making 
 the weight of the oxygen atom equal to 16, which is cus- 
 tomary. The weights of rings of electrons of 3, 4, etc., 
 on this scale will be 
 
 p electrons 
 per ring 
 
 Weight of ring 
 
 2 
 3 
 4 
 
 1 
 
 1 .008 
 2.268 
 4.032 
 6.300 
 9.072 
 
 • (221) 
 
 Founded upon this suggestion an attempt has been 
 made to ascertain from the known atomic weights of the 
 
134 
 
 The Atom 
 
 Table Calculated with the Numbers in (222) 
 
 •■oOK 
 
 <'gd 
 
 1.008 
 4.00 
 6.94 
 91 
 
 II. o 
 
 12.00 
 
 14.01 
 
 16.00 
 
 19.0 
 
 20.2 
 
 23.00 
 
 24.32 
 
 27.1 
 
 28.3 
 
 31.04 
 
 32.06 
 35.46 
 39-88 
 
 39.10 
 
 40.07 
 
 44.1 
 48.1 
 
 $1 .0 
 
 $2.0 
 54-93 
 
 55.84 
 58.97 
 58.68 
 63-57 
 65.37 
 
 69,9 
 
 72.5 
 
 74.96 
 
 79.2 
 
 79,92 
 
 82.92 
 85.45 
 87-63 
 
 •So« 
 
 1 .008 
 
 3-99975 
 6.8893 
 
 9-1371 
 11. 0235 
 
 11.99926 
 
 14.01587 
 
 15.9990 
 
 19.023 
 
 20.3115 
 
 23 . 0228 
 24.3112 
 27 . 0225 
 28.3109 
 31.0223 
 
 32.0625 
 
 35.4488 
 
 39-8953 
 39.054 
 40 . 062 
 
 44 . 062 
 48 . 062 
 
 51-053 
 52.061 
 54.919 
 
 55-831 
 59.020 
 58.638 
 63.613 
 65-317 
 70.012 
 
 72.589 
 75 . 020 
 79.185 
 79.905 
 
 82.917 
 85.494 
 87.612 
 
 ■3 
 
 ° o 
 
 .5" 
 
 O it 
 
 H 
 
 2 
 
 4 
 
 9 
 
 6 
 
 14 
 
 12 
 16 
 16 
 22 
 23 
 26 
 27 
 30 
 3« 
 34 
 
 40 
 29 
 
 43 
 46 
 48 
 
 52 
 56 
 58 
 60 
 61 
 
 38 
 62 
 66 
 68 
 69 
 
 72 
 74 
 78 
 80 
 68 
 
 89 
 91 
 92 
 
 Arrangement in 
 lings, in atoms 
 
 2 
 
 3 
 
 ♦ 
 
 5 
 
 6 
 
 T 
 
 
 
 3 
 
 I 
 
 
 
 3 
 2 
 
 3 
 
 2 
 
 3 
 
 2 
 
 3 
 
 2 
 
 3 
 
 8 
 
 I 
 
 7 
 8 
 
 8 
 8 
 
 I 
 
 4 
 
 T 
 
 3 
 3 
 
 2 
 
 3 
 3 
 
 4 
 4 
 4 
 
 5 
 
 6 
 
 7 
 
 6 
 
 I 
 8 
 8 
 8 
 
 9 
 10 
 II 
 11 
 II 
 
 
 
 6 
 
 1 
 3 
 3 
 
 2 
 
 3 
 2 
 
 I 
 
 4 
 
 2 
 
 3 
 
 2 
 
 2 
 I 
 
 2 
 2 
 
 3 
 
 5 
 
 2 
 
 14 
 12 
 14 
 15 
 
 17 
 17 
 18 
 
 15 
 
 4 
 
 18 
 18 
 20 
 
 2 
 10 
 
 ** [I 3 ? 
 
 "si- 
 
 0.0993 
 0.25 
 
 o. 144 
 10 
 
 909 
 0833 
 
 0714 
 
 0625 
 526 
 
 0.495 
 
 0.0435 
 0.0413 
 0.369 
 
 0-3533 
 0.0322 
 
 0.0312 
 0.0282 
 o . 025 I 
 .0256 
 o . 02495 
 
 0.227 
 0.208 
 
 . 196 
 
 . 192 
 0.0182 
 
 0.0179 
 0.0169 
 0.01705 
 
 0.01573 
 0.0153 
 
 o. 1431 
 
 0-1379 
 0133 
 
 1262 
 0125 
 
 01206 
 01 169 
 0II4 
 
 iS-S-S* 
 
 000 
 00616 
 
 731 
 408 
 214 
 
 00616 
 
 0376 
 
 00616 
 
 121 
 
 552 
 
 0.099 
 • 0.0362 
 0.286 
 o . 0387 
 0.0571 
 
 00793 
 0315 
 0384 
 1 175 
 0199 
 
 0867 
 
 0800 
 
 1045 
 1 178 
 
 0208 
 
 0.0165 
 0.0857 
 0.0717 
 0.0683 
 0.0814 
 
 o. 
 o. 
 o. 
 o, 
 o 
 
 1599 
 1223 
 0795 
 
 0192 
 
 0185 
 
 00378 
 
 0512 
 
 0206 
 
The Atom 
 
 135 
 
 ^-^ 1 
 ^£■2 3 
 
 «8.7 
 90.6 
 
 93- 1 
 
 96.0 
 
 101.7 
 
 102.9 
 
 106.7 
 
 107.88 
 112.40 
 114. 8 
 118. 7 
 120.2 
 
 127.5 
 
 126.92 
 
 130-2 
 
 132.81 
 
 137-37 
 
 139.0 
 
 181. 5 
 184.0 
 190.9 
 
 193- 1 
 195.2 
 
 197.2 
 200.6 
 204.0 
 207 . 20 
 208.0 
 
 222.4 
 226.0 
 232.4 
 238.2 
 
 
 88.900 
 
 90.604 
 
 93.181 
 
 96 . 026 
 101.595 
 
 102.883 
 
 106.603 
 
 107.891 
 112.306 
 
 114.883 
 
 118.602 
 120.305 
 
 127.456 
 126.938 
 
 130.288 
 132-795 
 137-312 
 
 139-015 
 
 181.590 
 184.021 
 
 190.878 
 
 193-174 
 195.293 
 
 197.308 
 
 200.613 
 
 204 . 020 
 207.292 
 208.052 
 
 222.315 
 226 . 034 
 
 232-331 
 238 . 282 
 
 
 93 
 
 94 
 
 96 
 100 
 104 
 105 
 no 
 
 III 
 115 
 117 
 122 
 123 
 
 125 
 109 
 
 131 
 140 
 141 
 
 Arrangement in 
 rings, in atoms. 
 
 142 3 34 
 
 Rare Earths 
 
 184 
 188 
 
 193 
 196 
 197 
 
 201 
 206 
 208 
 209 
 216 
 
 227 
 232 
 239 
 239 
 
 5 6 
 
 I 
 
 2 
 
 44 
 
 4 
 
 
 45 
 
 
 3 
 
 46 
 
 
 4 
 
 46 
 
 I 
 
 I 
 
 48 
 
 3 
 
 I 
 
 48 
 
 4 
 
 2 
 
 48 
 
 4 
 
 
 50 
 
 I 
 
 I 
 
 51 
 
 8 
 
 
 50 
 
 4 
 
 I 
 
 54 
 
 6 
 
 
 55 
 
 6 
 
 I 
 
 56 
 
 
 I 
 
 59 
 
 a € V .' 
 Ph " g " 
 
 O. II27 
 
 O.I 104 
 
 o. 1074 
 o. 1042 
 0.0983 
 o . 0972 
 0.09375 
 
 o . 00927 
 0.00889 
 0.0871 
 o . 0842 
 0.0832 
 
 o . 0784 
 0.00788 
 0.0768 
 0.00753 
 o . 00728 
 
 0.0719 
 
 0.0551 
 0.05435 
 05238 
 0.0518 
 0.0512 
 
 0.0507 
 0.04985 
 0.0490 
 o . 00483 
 0.04806 
 
 o . 04495 
 0.0442 
 o . 0430 
 0.0420 
 
 
 .226 
 . 00407 
 
 .0866 
 
 .0275 
 
 1033 
 
 .0162 
 .0912 
 
 0103 
 0838 
 
 0720 
 
 0826 
 0876 
 
 0346 
 0145 
 ,0680 
 
 Olio 
 
 0420 
 
 o . 0496 
 O.0II6 
 0.0II5 
 0.0385 
 o . 0474 
 
 0.0550 
 o . 00650 
 o . 00965 
 0.0443 
 o . 0249 
 
 •0381 
 .0152 
 .0298 
 
 .0344 
 
 (223) 
 
136 
 
 The Atom 
 
 difFerent kinds of atoms the particular combination of 
 rings of electrons, each having these approximate weights, 
 as given in the table, which will add up to make the 
 known weight of the atom. After many trials it has been 
 found that, by sHghtly altering the numbers in the table 
 according to the following values, a very large majority 
 of atomic weights may be obtained with accuracy, namely 
 
 p electrons 
 per ring 
 
 Weight of ring 
 
 2 
 3 
 
 4 
 
 1.008 
 2.29643 
 
 9- 137 
 
 . . (222) 
 
 It will make the process of calculation clearer if an 
 example is given. The element magnesium is given in 
 the table as having eight rings of electrons total made 
 up of five rings of four, one ring of three, and two rings 
 of two electrons, or a total of 27 electrons. The weights 
 supposed to be contributed to the atom by the rings are 
 as follows: 
 
 5 rings of four = 5 x 3-99975 = 19.99875 
 
 1 ring of three = i x 2.29643 = 2.29643 
 
 2 rings of two = 2 x 1.008 = 2.016 
 
 Total 24.3112 (224) 
 
 The measured atomic weight of magnesium is 24.32, 
 and admitting that it is possible that an error of at least 
 one unit in the last decimal place in the experimental 
 weight has been made, the calculated value comes within 
 the experimental error. 
 
 It may be urged as an objection to this scheme that 
 the number of possible combinations which can be made 
 
The Atom 137 
 
 of these numbers that will approximate to any desired 
 number is very great. This objection has some force, 
 but it will not apply to the elements of low atomic weight, 
 and it is here that a real test of the scheme is obtained. 
 The calculation has been extended to the heavier ele- 
 ments largely because the first part of the table of low 
 atomic weights indicates the nature of the whole scheme 
 as being largely based upon a preponderating number of 
 rings of four electrons, and scattering numbers of rings 
 of three and of two. Rings of five and six are seldom 
 required, and no ring greater than six appears. Rings 
 of five appear in the halogens, chlorine having five, 
 bromine ten, and iodine fifteen. 
 
 It is not contended that the above table gives the 
 correct number of electrons in the atoms in every case, 
 and will not be subject to future revision, but the main 
 idea which runs through the table is that most of the 
 atoms are made up of rings of four electrons. Nothing 
 is intimated as to the order of succession of the rings as 
 we proceed out from the nucleus, that is, whether the 
 rings of two are within or outside of the rings of three or 
 four. If we take one atom of each kind as given in the 
 table and add up the number of rings of electrons to 
 find a total, we shall find that there are in the 70 atoms 
 
 1470 rings of four electrons 
 185 rings of two electrons 
 86 rings of three electrons 
 35 rings of five electrons ^ 
 
 7 rings of six electrons (225) 
 
 thus indicating that the rings of four electrons greatly 
 predominate in any mass of matter made up of a mixture 
 of elements, such as the earth for example. 
 
 This feature which is essential to the scheme may 
 
138 The Atom 
 
 be tested by means of the gravitational equation, as will 
 now be explained, and it is proved to be in complete 
 harmony with the gravitational equation. 
 
 As a first illustration, let us apply the formula to 
 obtain the attraction between the earth and a single 
 hydrogen atom upon its surface. We have, by Newton's 
 law, 
 
 F = kmHrnsrE'-^ ... . (226) 
 
 and, by the gravitational equation, 
 
 F = ig^moSiS^S/SVB-^ (227) 
 
 H E 
 
 Equating these two expressions for the same force, it is 
 seen that the only unknown quantity is 2/3^ for the 
 
 E 
 
 earth, which may, therefore, be found. It is 
 
 Substituting the following numerical values, namely 
 
 k = 666x lo-i" 
 rriH = 1.662 X io~^ 
 rriE = 5.984 X 10^^ 
 
 e = 4.774 X 10-'" 
 mo = .90 X io~2' 
 
 l*^^ ^ ^^^^^ (^°7)) = -531 X 10-^. 
 We obtain numerically 
 
 S/32 = 1.825 X 10*^ (229) 
 
 Use may be made of this sum of the squares of the 
 velocities of all the electrons in the earth to find the 
 average speed of a single electron in the earth by divid- 
 ing this number by the total number of electron? in the 
 
The Atom 
 
 139 
 
 earth. Fortunately we know the approximate number 
 of electrons in the earth as nearly as we know the mass of 
 the earth in grams, for the number of electrons per gram 
 of all substances except hydrogen is equal to the Avogadro 
 constant, namely, 6.062 X 10*'- The total number of 
 electrons in the earth is then approximately 
 
 N = 6.062 X lo^'m^ = 3.6275 X 10^'. , . (230) 
 
 And now dividing S/3'' fay N we obtain the mean square 
 
 E 
 
 velocity of a single electron in the earth, as 
 
 /8/ = -503 X 10-", (231) 
 
 and taking the square root, the average velocity of an 
 electron in the earth is 
 
 % = .0071 (232) 
 
 For comparison, we give the numerical values of the 
 velocities of the electrons in rings of electrons calculated 
 from the formula (170). 
 
 p electrons 
 per ring 
 
 jS = the velocity in 
 terms of ve- 
 locity of light 
 
 2 
 3 
 
 4 
 
 i 
 
 0.003641 
 0.005462 
 0.007283 
 0.009104 
 0.010925 
 
 (233) 
 
 It is seen from this table that the velocity of an electron 
 in a ring of four electrons is very nearly equal to and a 
 little greater than the average velocity of an electron in 
 the earth, the comparison being .0071 and .00728. The 
 average velocity for the earth is much nearer to that of 
 a ring of four than to that of any other ring. This sup- 
 ports in a very striking way the scheme in the atomic 
 
140 The Atom 
 
 weight table above given and shows that the number of 
 rings of four electrons in the atoms of the earth greatly 
 preponderate, as it should according to the table. 
 
 Now it will be noticed that the mass of the earth 
 occurs both in (228) and (230), so that when one is divided 
 by the other this mass canceled. No error was intro- 
 duced into the result, therefore, because of a wrong 
 value for the mass of the earth. This leads to important 
 considerations because it is apparent that for the earth 
 might have been substituted any other body of mixed 
 matter, the sun or any of the planets, and we would have 
 obtained the same average velocity for a single electron. 
 This does not seem strange because, according to the 
 scheme of the atomic weight table, the average velocity 
 should be nearly equal to that in a ring of four electrons. 
 But let us look at the equations with more attention 
 because of this general result. 
 
 Let the gravitational equation be applied to the hy- 
 pothetical case of the attraction between a star or a 
 mass of nebulous gas composed, first, entirely of hydrogen 
 and a single hydrogen atom at a great distance away, 
 somewhere outside of the star. If M denotes the mass 
 of the star, the Newtonian law gives the attraction as 
 
 F = kmnMr-^, (234) 
 
 and it does not matter what value r has, provided the 
 single hydrogen atom with mass mn is situated in a fixed 
 position a long distance from the star. The general ex- 
 pression for mass is given in (219) above, and the ratio 
 of these two masses is, therefore, 
 
 M/iriH = S,8V2/32 (235) 
 
 MB 
 
 But the sum of j8s^ for a single hydrogen atom is twice 
 the square of the velocity of just one of the electrons 
 
The Atom 141 
 
 because there are two electrons, and we will denote this 
 velocity by j8s. The last equation is then equivalent to 
 
 /3«2=^- (236) 
 
 — M 
 
 Now the average speed of the electrons in the star must 
 be the same as a single electron in an hydrogen atom, 
 Pb, because we have made the hypothesis that the star 
 is entirely composed of hydrogen atoms. And since the 
 numerator on the right of the equation represents the 
 sum of the squares of the velocities of all the electrons 
 in the star, and the quotient, jSu^ the left member of the 
 equation, represents the average square of the velocities, 
 it must be that the denominator on the right represents 
 the total number of the electrons in the star, say N, 
 
 N = — M (237) 
 
 And since M is the number of grams mass of the star, 
 then 2/mB must be the number of electrons in one gram 
 mass of hydrogen. This is evidently true because vhh 
 represents the mass of one hydrogen atom, and the number 
 of such atoms in one gram must be i/rriH. The number 
 of electrons per gram is twice this quantity, or i/niH. 
 The reciprocal of the mass of the hydrogen atom i/nia 
 is very nearly equal to the Avogadro constant, and we 
 may say, then, that the number of electrons in a gram of 
 hydrogen is about twice the Avogadro constant. The 
 number of electrons in any other element, however, is 
 nearly equal to the Avogadro constant, and not twice it, 
 as we shall see, hydrogen being an exception, as it is in 
 several other particulars. 
 
 Let us now take another examplq and suppose that 
 
142 The Atom 
 
 the star is entirely composed of helium instead of hydro- 
 gen, and assume that the hehum atom has a single ring 
 of four electrons. The ratio of the masses of the star, 
 M, to the single hehum atom, m^e is then 
 
 M/rriH, = -L^J-L^^ (238) 
 
 M Be 
 
 Denoting the speed of one electron in the helium atom 
 by ^hb, then 
 
 ^fi' = 4^He (239) 
 
 He 
 
 and (238) becomes equivalent to 
 
 2)82 
 
 PHe' = -^ (240) 
 
 mHe 
 
 In a similar manner to the case of the hydrogen star, 
 the numerator on the right represents the sum of the 
 squares of the velocities of all the electrons in the star, 
 and the quotient on the left represents the average square 
 of the velocity of one electron in the star. Hence the 
 denominator on the right must represent the total number 
 of electrons in the star, say N, 
 
 N = ^M. (241) 
 
 rriHe ^ ^ ' 
 
 Hence /^/mne must be the number of electrons per gram 
 of helium. And evidently i/rriHe is the number of hehum 
 atoms in one gram, and four times this is the number of 
 electrons in the gram. 
 
 The mass of the hehum atom in terms of that of hydro- 
 gen is 
 
 mHe = jT^ rrin = 3-96837713 .... (242) 
 
 and 
 
 ^/rtiHe = 1-00%/mH (243) 
 
The Atom 143 
 
 Using the Millikan value of ma = 1.662 x io~^S we obtain 
 
 4/mHe = 6.065 X 10^', the number of 
 
 electrons in one gram of helium. . . . (244) 
 
 This number is very approximately equal to the well- 
 known Avogadro constant, which is given by Millikan as 
 6.062 X 10^'. In deriving his result Millikan used a 
 little more accurate value of the atomic weight of hydro- 
 gen, namely 1.0077 instead of 1.008, which would account 
 for the difference in the last decimal place of the Avogadro 
 constant. It thus appears that the number of electrons 
 per gram of helium is equal to the Avogadro constant, 
 and not twice this number as in the case of hydrogen. 
 The average velocity of the electrons in this helium star 
 is also the same as the velocity in a ring of four electrons, 
 and approximately the same as the average velocity of the 
 electrons in any other piece of matter made up of a 
 mixture of different kinds of atoms. 
 
 This is a direct result of the fact that the number of 
 electrons per gram of other substances than hydrogen is 
 very nearly constant and equal to the Avogadro constant, 
 and the reason for this is to be found in the scheme of the 
 atomic weight table, which makes the rings of four elec- 
 trons greatly preponderate over the other kinds of rings. 
 It may be proved that the number of electrons per gram 
 of all substances other than hydrogen is approximately 
 constant, whether we assume that the number of elec- 
 trons per a.tom is proportional either to the atomic num- 
 ber or to the atomic weight, it does not matter which. 
 But instead of digressing here to give this proof, it is 
 added in Appendix B. There has been considerable 
 controversy over the question of the number of electrons 
 in the various kinds of atoms, and some have made the 
 number roughly proportional to the atomic number, 
 
144 ^^^ Atom 
 
 others to the atomic weight. Admitting that the hydro- 
 gen atom has two instead of one electron reconciles these 
 two points of view, for the ratio of the number of elec- 
 trons in an atom to the number in hydrogen may be 
 considered to be very near to the atomic number, whereas 
 the actual number may be nearer to the atomic weight, 
 which is roughly twice the atomic number. There is 
 some ground, therefore, for both contentions, or at least 
 it is possible to see how both views have been held. 
 
 It is probable that one of the principal objections that 
 will be raised to the atomic weight table (223) will be 
 that the number of electrons in the various atoms is not 
 made equal to the atomic number. It is the common 
 idea among writers on these subjects to think of the 
 number of positive unit charges on the nucleus as equal 
 to the atomic number, that of hydrogen being unity, 
 helium two, etc., exactly according to the atomic number. 
 This conception has come about through the work of 
 Moseley, who first obtained a linear relation between the 
 different atoms by means of the X-ray spectra, and it 
 was natural to infer that the uniform progression of the 
 X-ray frequencies is connected with the fact that the 
 charge on the electron has a fixed value and that their 
 number increases by unity from element to element. 
 
 In reply to this it may be pointed out that this is still 
 an inference not supported by any conclusive proof. 
 The present theory seems to throw some light on this 
 matter, for it has been rendered probable that the spectra 
 of atoms is due primarily to the motion of the electrons 
 connected with it, and we should look to this for regu- 
 larities in the X-ray or the light spectra rather than 
 directly to the nucleus. The quantity, b, Planck's con- 
 stant, plays a part in this, and we have in this constant 
 just as truly a fixed value as the value of the electrical 
 
The Atom 145 
 
 charge, and which might as easily be responsible for a 
 regular increase in the spectra from element to element 
 as electrical charge. 
 
 One might be tempted to suppose that the mass of the 
 nucleus should also proceed by regular steps if its electrical 
 charge increases by equal steps, but it does not. It may 
 be worth while to see how the mass of the iron atom, for 
 example, can be so great as to have atomic weight 55.84 
 and yet have the comparatively small number of electrons 
 allotted to it in the table (38) in the light of this theory. 
 The mass of the nucleus depends as well upon its radius 
 as it does upon its electrical charge, according to the 
 Lorentz mass formula. In the case of hydrogen we have 
 seen that there is a direct relation between its radius and 
 the Rydberg constant, namely 
 
 as = S/sKk (See (161).) 
 
 And this again is connected with the period of revolution 
 of the electrons about the nucleus, the frequency being 
 equal to 2K, for the ring of two electrons. In the case 
 of the iron atom, which we have made to consist of six 
 rings of six and one ring of two electrons, the principal 
 part of the frequency is that of a ring of six electrons 
 instead of two. It seems legitimate to say that the 
 radius of the iron nucleus is connected with the Rydberg 
 constant by an analogous expression to that in (161), 
 but that, the frequency of the rings of six being greater, 
 we should have the radius, a, proportionately smaller 
 than is the case in hydrogen. This would make the mass 
 of the atom, which is inversely as its radius, proportion- 
 ately larger than would correspond to the same number 
 of electrons if they had the same speed as in hydrogen. 
 In other words, it is easy to see by the aid of this theory 
 that the mass of the iron atom may be 55.84, while the 
 
146 The Atom 
 
 number of electrons is exceptionally low, since the speeds 
 or frequencies of revolution are exceptionally high. It 
 is yet too early to give any exact calculation based upon 
 the theory in the complicated case of iron of the radius 
 of its nucleus. Its spectrum should first be reduced to a 
 definite formula in a similar manner to that of hydrogen. 
 It appears to the author to be too simple a solution 
 of an involved question to assume, as is commonly done, 
 that the charge of the atomic nucleus increases by a 
 fixed value from element to element in exact agreement 
 with the atomic number. 
 
APPENDIX A 
 
 It is proposed to show i that the force obtained from the 
 gravitational equation (201) above is such that the 
 bracket in that equation may be replaced by the numeric 
 f when the orbits of the two electrons take their average 
 position or orientation with respect to each other in space, 
 without, of course, altering the distance between their 
 centers, r. 
 
 The quantities within the bracket, when expanded, 
 become 
 
 I - X^ sin* a + 2XZ sin a cos a - Z* cos* a, . (i) 
 
 and we always have the relation between the direction 
 cosines as follows, 
 
 X2 + 72 + Z* = I (2) 
 
 First, considering the center of the orbit of 62 as fixed in 
 position in space, the quantities r and z or Z are fixed, 
 but not X and y, or X and Y, due to the way in which the 
 axes have been defined. Let us now suppose that a is 
 fixed and that the pole of the orbit of ej rotates around 
 the elementary small circle of the sphere at a fixed lati- 
 tude through an arbitrary angle, <^, from zero to 27r. It 
 is easy to show from the definition of the axes that 
 
 , X X X . , 
 
 ^°^*'* " (x* +y*)» " (r* - z*)* ~ (I - Z*)i • • • ^3; 
 
 The value of the bracket (i) then becomes in terms of 
 
 I - (i - Z*) sin* a cos* <^ 
 + 2 sin a cos a(\ - Z^YZ cos <^ - Z* cos* a (4) 
 
 ' hoc. cit., Pbys. Rev., July, 1918, pp. 19, 20, and 21. 
 
 147 
 
148 The Atom 
 
 Regarding all quantities in this except ^ as constant, 
 we may obtain the average by integrating between zero 
 and 2ir. The average of cos^ </> is |, and of cos ^ is zero 
 between these limits, giving the result, 
 
 I - |(i - Z^) sin" a-Z'^ cos^ a, .... (5) 
 
 which is equivalent to 
 
 Ki + Z=) + Ki - 3Z') cos" a (6) 
 
 Let us next assume that Z remains fixed and average 
 for a, thus obtaining the average over the first whole 
 sphere. Again replace cos* a by its average |, giving 
 
 Ki+Z=)+Ki-3Z^)=f-|Z^- • • • (7) 
 
 And finally average for a change of Z between the 
 limits zero and unity, as the center of the system of 
 orbits, 62, moves around the center of Ci at the fixed radius, 
 r, from the equator to the pole. The average of Z^ be- 
 tween these limits is |, so that (7) becomes 
 
 1-1^ = 1 (8) 
 
 This completes the proof that the bracket in the gravi- 
 tational equation (201) or (i) may be replaced by the 
 numeric f on the assumption that it is equally probable 
 that the pole of the orbit of each electron ei and d will 
 lie in any one unit area of a sphere surrounding each 
 center as in any other unit area of these spheres. This 
 must be the case generally in all kinds of matter, — 
 solids, liquids or gases, — crystals only excepted. 
 
 We will next show that this value | will probably be 
 obtained in the case of crystals also. To do this pre- 
 supposes that the directions of the axes of rotation of 
 the atoms in the crystal are known. In the cubic or 
 isometric system of crystals, which is the only system 
 
The Atom 149 
 
 that the author has theoretically investigated' as yet, 
 it has been proved that the directions of all the axes of 
 rotation are equally divided into four groups, the axes 
 in each group being all parallel to each other. The 
 relative directions of these axes as between the four dif- 
 ferent groups are exactly according to the relative direc- 
 tions of the four medial lines of a regular tetrahedron. 
 That is to say, one group has axes parallel to one of the 
 medial lines of the tetrahedron, the second group to the 
 second medial line, the third to the third, and the fourth 
 to the fourth. 
 
 The proof of this proposition is very simple and depends 
 only upon the fact that each atom exerts a turning mo- 
 ment of force upon every other atom in the crystal tend- 
 ing to turn the plane of its orbit until it comes into 
 parallelism with the given atom, when the turning mo- 
 ment vanishes. Without knowing or assuming any law 
 governing these turning moments, if we merely assume 
 that these moments are the same in similar positions of 
 the atoms, then it may easily be shown that the sum of 
 the turning moments of all the other atoms in the crystal 
 acting upon any selected atom becomes zero and also 
 produces stability, so that any displacement from this 
 stable position brings into play a restoring couple, pro- 
 vided the directions of all the axes of rotation are grouped 
 in the way that has been published elsewhere. This 
 grouping makes the axes of all atoms take directions in 
 four equal groups parallel to the four medial lines of a 
 regular tetrahedron, as just stated. 
 
 To study the behavior of a cubic crystal, therefore, 
 
 ' Loc. cU., Pbil. Mag., June, 1915, p. 750, particularly p. 763, and 
 Figure I. A. C. Crehore, Pbil. Mag., Vol. XXX, August, 1915, 
 p. 257. 
 
 ' Loc. cit., Pbil. Mag., June, 1915, pp. 766, 767. 
 
150 The Atom 
 
 by means of the gravitational equation, it is only neces- 
 sary to study the behavior of a group of four electrons, 
 the orbits of which have their axes parallel to the four 
 medial lines of a regular tetrahedron respectively. First, 
 it may easily be shown that the quantity 
 
 r* - (- X sin a + z cos a)^ 
 
 is geometrically represented by the square of the per- 
 pendicular line from the center of the orbit of the electron, 
 Ci, upon which we are getting the force, drawn to the 
 axis of rotation of the electron ej. Dividing this expres- 
 sion through by r^ gives 
 
 I - (- X sina + Z cos a)*, 
 
 which occurs within the bracket of equation (201). If, 
 now, we write down the force according to (201) for each 
 of the four electrons, d, having their axes parallel to the 
 four medial lines of a regular tetrahedron respectively, 
 and add together these four forces to obtain the effect 
 of the group of four upon the one electron, Ci, we are in 
 effect adding together the squares of the four perpen- 
 dicular lines drawn from the center of Ci upon each of the 
 four axes of the orbits of 62. 
 
 The sum of these forces will be a constant quantity, 
 no matter in what direction the tetrahedron is turned, 
 for it has been possible to establish the truth of the fol- 
 lowing geometrical theorem * upon which this matter de- 
 pends. " If through any point four lines be drawn, 
 making equal angles each with any other, and if from a 
 second point at a fixed distance, r, from the first point 
 four perpendiculars be drawn one to each of the said 
 four lines, then the sum of the squares of these perpen- 
 diculars is constant for all points at the same distance 
 
 ' Loc. cit., Pbys. Rev., June, 1917, p. 459. 
 
The Atom 151 
 
 from the first point. The locus of the second point is 
 the surface of a sphere with the first point as center." 
 
 It can make no difference in the force upon the electron 
 Ci, therefore, how the group of four electrons is oriented 
 with respect to it. This is true of any other electron in 
 the body of which ci is a selected electron. It is also 
 true of a multitude of groups of four electrons, Cz, in the 
 body 2, which may be supposed to be a cubic crystal. 
 Hence we conclude that the gravitational equation shows 
 that the attraction between two cubic crystals is strictly 
 independent of their relative orientation, as it is known 
 to be in fact. 
 
 It remains to obtain the value of the bracket in equa- 
 tion (201), and see whether it can be replaced by the 
 factor f as it was in the case of the general average ob- 
 tained for all solids, liquids, and gases above. Since it 
 makes no difference how the single group of four electrons 
 is oriented, let us so place the medial lines of the tetra- 
 hedron that one of them passes directly through the 
 centers of the orbits of Ci and of e^. The other three 
 medial lines will then make equal angles with the line 
 joining centers. The perpendicular distance from the 
 center of the orbit of Ci upon each of these other three 
 
 medial lines of the tetrahedron is then equal to r, 
 
 3 
 
 Q 
 
 and the sum of their squares is - r^. Hence the sum of the 
 
 four forces is given by replacing the bracket in (201) by 
 the quantity |, and consequently the average force per 
 pair of electrons as the group is oriented in all possible 
 ways is |th of this, or the bracket must be replaced by 
 the quantity f, which is precisely the same value that 
 was obtained above for any other kind of substance. 
 In advance of the theoretical investigation of other 
 
152 The Atom 
 
 systems of crystals it cannot be said that we have es- 
 tablished this proposition in general for all kinds of 
 crystals. But it will certainly be a good guide in the 
 study of other crystals first to assume that all axes are 
 divided into four equal groups, each parallel to one of the 
 medial lines of a regular tetrahedron, and see whether 
 this proposition cannot be generally established. 
 
APPENDIX B 
 
 Prove that the number of electrons in a gram ' of sub- 
 stances in general, hydrogen excepted, is a constant 
 quantity, if we start with the assumption that the num- 
 ber of electrons per atom is proportional either to the 
 atomic number or to the atomic weight. 
 
 It is known that in a perfect gas, whether elemental or 
 compound, the number of molecules per cubic centi- 
 meter is a constant quantity under standard conditions 
 of pressure and temperature. This number is referred 
 to as the gas-constant and may be denoted by N, say. 
 If d is the density of the gas, the mass of a volume, V, 
 of it is m = W. If we confine the attention to a volume 
 of gas which has a mass of one gram, m = i and V = \/d. 
 The number of molecules in one cubic centimeter is N, 
 and the molecules in one gram are, therefore, NV = N/d. 
 
 Let us now assume that there are p electrons in one 
 molecule of the gas. Then the number of electrons per 
 gram must be equal to 
 
 A = pNV = pN/d. 
 
 For two different gases it may be shown that p/p' = d/d', 
 or p/d = p'/d', on the hypothesis that p is proportional 
 to the atomic weight or to the atomic number, and hence 
 A = pN/d = p'N/d' = p"N/d" = a constant quantity. 
 
 It remains to prove that p/d = p'/d' = p"/d", etc., 
 for different gases. Suppose that the complex molecule 
 of the gas is made up of n\ atoms having atomic weight 
 
 » Ai C. Crehore. Phys. Rev., N. S., Vol. X, No. 5, October, 1917. 
 See pp. 447, 448. 
 
 153 
 
154 The Atom 
 
 Ai, 7i2 atoms having atomic weight Ai, etc., then the 
 weight of the molecule, M, the molecular weight, is 
 
 M = riiAi + UiAt + nzAz + • • • etc. 
 
 If the number of electrons per atom, P, is proportional 
 to the atomic number or to the atomic weight, we have 
 the atomic weights, Ai, Ai, etc., equal to some constant, 
 say b, times their respective numbers of electrons. Pi, 
 Pi, etc., that is 
 
 Ai = bPi; Ai = hPi, etc. 
 Hence 
 
 M = 6(niPi + UiPi + etc.) = hp, 
 
 since the number of electrons per molecule, p, is equal to 
 
 p = riiPi + nJPi + etc. 
 
 Multiplying the molecular weight, M, by the number of 
 molecules in one cubic centimeter of the gas, N, gives 
 the mass contained in one cubic centimeter of it, as 
 
 NM = Nhp. 
 
 But the mass per cubic centimeter of a gas is the density 
 of the gas by definition, hence 
 
 d = Nbp, 
 
 where A^ and 6 do not vary for different gases. For 
 another gas this becomes 
 
 d' = Nbp', 
 
 whence p/d = p'/d' = p"/d" = etc. = ifNb, a constant. 
 This establishes the proposition proposed, and makes 
 the number of electrons per gram 
 
 A = pN/d = i/6, a constant. 
 
 The constant A may be considered as equal to the Avo- 
 gadro constant, 6.062 X 10^, and the constant 6 as the 
 reciprocal of this. 
 
=(t)' 
 
 APPENDIX C 
 
 Guided by the new space-time system of dimensions, 
 it has been attempted to discover some function of the 
 fundamental constants that will represent Planck's con- 
 stant, h, both in numerical value and in dimensions, for 
 there has appeared no place in the work above from which 
 this important constant might be obtained except in the 
 single instance of the Newtonian constant. It is con- 
 sidered that this is not known experimentally with an 
 accuracy suflScient for our purposes. It will prove of 
 interest to give the several forms of expressions for b 
 thus found, which are equivalent. They are, first, 
 
 if'. C) 
 
 where aa = the radius of the hydrogen nucleus, 2K is 
 the frequency of revolution of the electrons in the normal 
 hydrogen atom equal to twice the Rydberg constant, 
 and c the velocity of light. 
 
 The dimensions of h must be those of energy multiplied 
 by a time. On the electrostatic system energy has the 
 dimensions UMT'^W, and putting mass equal to a ve- 
 locity, we obtain for energy L^T~\ or the cube of a velocity. 
 Multiplying these by a time, the dimensions of b in the 
 electrostatic system are L^MT~^}^, and on the space- 
 time system L^T~^. 
 
 The dimensions of the above expression for b are, 
 therefore, in agreement with the required dimensions on 
 the space-time system, for ag^/c = LT, and K^ = T~\ 
 whence qh^K^/c = LT"^. 
 
 155 
 
156 The Atom 
 
 In (161) above we have given a value of the radius of 
 the hydrogen nucleus in terms of the Rydberg constant, 
 
 namely an = S/^Kk (2) 
 
 By means of this we may eliminate an from (i) and 
 find 
 
 h = 8yi3*k*Kc (3) 
 
 The dimensions of this expression for b are the same 
 as those of (i) if we regard k as the reciprocal of a velocity, 
 foi- i/k* = UT-*, and i/Kc = L-'T^, whence i/k*Kc = 
 L^T~^, the dimensions of h. This expression makes 
 b depend upon accurately known constants, K and c. . 
 The numerical value of k is unity, and 
 
 i/ifc=i/3.290xio'*X3Xioi''=io.i3i7ixio-" .... (4) 
 Also 8V15* = 32768/50625 = 0.647269. ... (5) 
 
 Multiplying these together, we obtain as the numerical 
 value of b, 
 
 b = 6.5579 X 10-" (6) 
 
 The velocity of light has been taken as the even num- 
 ber 3 X 10^"; the Rydberg constant as the even number 
 3.29 X 10'*, and the decimal places have been retained 
 on this account. When the best values of these two 
 constants are employed the velocity ot light will be a 
 very little smaller, making b a very little larger. It may 
 be remarked that the value just obtained is very close 
 to the value of b, namely 6.56 X lo"''', which was ob- 
 tained by Millikan as the best figure representing the 
 total result of his experiments on the emission of electrons 
 from fresh metallic surfaces in vacuo by light of different 
 frequencies. This was his machine-shop-in-vacuo ex- 
 periment, carried out primarily to verify the Einstein 
 equation that makes energy proportional to frequency. 
 This experiment resulted in an unusually good straight 
 
«-=-^^(?J' (7) 
 
 The Atom 157 
 
 line, all the observed points lying close to the line through- 
 out the whole range of frequencies observed, and this 
 established the proportionality in a very satisfactory 
 manner. The slope of the line gave the ratio of e to b, 
 whence b was calculated from the previous knowledge 
 of e. 
 
 And again, by means of the Lorentz mass formula, 
 given in (160) above, namely 
 
 4. 
 
 we may eliminate an from equation (i) and find 
 
 \15mflfec7 c 
 And by means of the expression for 2K in (149), namely 
 
 2K = ma y j , (9) 
 
 we find another expression for b : 
 
 \i5k) rriHC^ 
 
 The dimensions of this are correct, for the denominator 
 is dimensionless, since ttihc^ is the fourth power of a 
 velocity and fe* is the reciprocal of the fourth power of 
 a velocity. This leaves the dimensions of h the same 
 as those of e^, as it should be according to the space-time 
 system, as pointed out above. 
 
 Numerically, if we use for e^/mg the value 1.36778x10*, 
 obtained in (154) above, and take c = 3 X lo^', we find 
 
 , / 16V 1-36778 X 10* I o 
 
 ^-(lifejX Vxio^ =feiX 1.294538 
 
 X 5.06585 X I0-" = 6.5579 X I0-", . . (11) 
 the same value found in (6). 
 
158 The Atom 
 
 Referring to the original equation (i), it is to be ob- 
 served that the specific inductive capacity, k, as well 
 as the masses of the electron and nucleus, are absent, 
 so that no uncertainty enters this expression because of 
 any possible doubt that specific inductive capacity is 
 the reciprocal of a velocity, and that mass is a velocity. 
 The quantity 2K appears as representing the frequency 
 of revolution of the electrons in the normal hydrogen 
 atom, and as possibly that of the nucleus itself. There 
 is no immediately assignable reason why the |d part of 
 the radius should appear in the expression instead of the 
 whole radius; but it is not unnatural that fd should be 
 required in connection with a spherical shape. The 
 volume of a conical portion of a sphere is fd the radius 
 times the area included by the base. 
 
 It seems as if many of these incomprehensible matters 
 might become more comprehensible if we did not have 
 to use such large units of length and of time as the centi- 
 meter and the second in measuring atomic quantities, 
 which are so small in comparison. Accordingly, let us 
 adopt as the unit of length the distance that light travels 
 while the electrons are making just one revolution in 
 the hydrogen atom, namely in a time 1/2K seconds. And 
 let us take the time of one revolution as a unit of time 
 instead of using the second. 
 
 To convert the several kinds of quantities that con- 
 tinually occur over into this new system, we have 
 A unit of length (new) = 3 X 10^"/ 2K = 4.559,27 x io~* cm. 
 A unit of time (new) = 1/2K = .151,975 X io~'* seconds. 
 One second (old) = 2K = 6.58 X 10** new units. 
 
 To obtain the new values of other quantities, involv- 
 ing powers of L or T or both in the dimensional formula, 
 multiply each L by 2K/3 x lo'*, and each T by zK. 
 For example, the velocity of light becomes 
 
The Atom 159 
 
 2K I 
 c = 3 X lo*" cm. per sec. (old) = 3 x lo^" 
 
 3 X 10^" 2K 
 = I unit = LT~\ 
 
 and is unity on the new system of units, as is evident 
 because light travels by hypothesis a unit distance in 
 the time of one revolution, or unit time. So any ve- 
 locity on the new system is numerically equal to that on 
 the old divided by 3 x lo"*. 
 
 Mass has the dimensions of a velocity. Hence one 
 gram on the old systerti becomes 0.333,333 X lo"!" units 
 on the new system, and the unit of mass on the new 
 system is 3 x lo"* grams, which would be represented by 
 a cube of water about 31.07 meters on an edge. This 
 unit of mass is thus inconveniently large for a practical 
 system of units. The mass of the hydrogen atom be- 
 comes 
 
 niH = 1.658 X 10-273 X iqI" 
 
 = 0.55267 X 10-'* units = LT~^. 
 
 And the mass of the electron becomes 
 
 mo'= .898 X ■i6-"/3 X iQi" = 0.29933 X io-»^ units. 
 
 Specific inductive capacity becomes 
 
 k = I (old system) = L-^T = 3 x 10", 
 
 and is numerically equal to the velocity of Ught on the 
 old system. 
 Twice the Rydberg constant becomes 
 
 zK = r-i = 6.58 X io»6 (old) 
 = 6.58x10"^ 
 = unity in the new system. 
 
i6o The Atom 
 
 The equation (149) for the Rydberg constant is 
 
 2K = m^Ql 
 
 and in this, since both 2K and c are numerically equal 
 to unity, it appears that mu is numerically equal to e\ 
 although not having the same dimensions. Hence on 
 the new system 
 
 e* = 0.55267 X 10-'*, and e = 0.743 X io~" units. 
 
 The equation (149), when written 
 
 Energy = 2Ke'^ = mac^ = L^T-\ 
 
 represents energy, namely the probable energy content 
 of the hydrogen nucleus given above (see page 131) as 
 1.492 X 10""* ergs. Since, however, 2K and c are now 
 unity, the mass of the nucleus is numerically equal to the 
 energy content of the nucleus when expressed in the new 
 unit of energy. The new unit of energy becomes 
 
 One new unit of energy = 27 x lo"" ergs, 
 
 and One erg = 3.703,703 x io~^ new units. 
 
 Since one joule is equal to 10' ergs, it is equivalent to 
 3-7037 X 10"^' new units of energy. The number of 
 nuclei in one gram of hydrogen is equal to the Avogadro 
 constant, 6.062 x 10". The total energy content of one 
 gram of hydrogen is then 
 
 6.062 X 10'* X 1.492 X 10-' = 9.0457 X 10'" ergs 
 
 = 9.0457 X 10" joules. 
 
 If it is now supposed that this energy may by some means 
 be extracted at the rate of 1000 watts, or i kilowatt con- 
 tinuously, the total supply in one gram of hydrogen will , 
 last at this rate for 
 
 9.0457 X lo'Viooo =^ 9.0457 X 10" seconds. 
 
The Atom i6i 
 
 This nuqiber of seconds is about equivalent to 2870 
 years' time. The practical unit has been used in this 
 calculation as being more familiar. We have a better 
 appreciation of the enormous amount of energy stored 
 in the gram of hydrogen when expressed in units easily 
 grasped. 
 
 In conclusion it seems worth remarking that Planck's 
 constant, b, takes the very simple numerical value 
 
 \iSk/ \I5 X 3 X 10"/ "■ ^'^^ 
 = 1.598 X iO-*2, 
 
 on the new system of units of length and time because 
 of the relation in (10) above, in which the factor e^/mnc^ 
 becomes numerically equal to unity because c = i, and 
 e^ = rnn numerically, k is numerically equal to the 
 velocity of light on the C.G.S. system. The dimensions 
 of the numerical expression for b just given are not com- 
 plete without the factor as in (10). This case is very 
 much like the common practice in the present' system of 
 units of suppressing the specific inductive capacity, 
 which alters the dimensions of the expressions. If we 
 had always used these new units instead of the centimeter 
 and the second, this factor e^/mnc^ might have been cus- 
 tomarily suppressed, and we should have a false idea of 
 the dimensions of h. 
 
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