RELATIONS BETWEEN FUNDAMENTAL PHYSICAL CONSTANTS BY J. E. MILLS
While engaged in an investigation of attractive forces the author calculated some functions involving basic physical constants which would not be calculated from the point of view usually adopted. It was found that certain similar numbers resulted in the most unexpected manner. For example:-
xmG 18861; C2
V2
=
188jo; 3xGc = 18829;-
€2
= 18818 1846.8h ~ / z x c = 0.j3088 X IO-”; e/m, = 0,53035 X 10‘~; a = 0.53084 X IO-*; 477 1846.8~= 42.859 X IO^; h2 = 42.863 X IO-^^; pv2 = 42.861 X IO-’’;
=
For the constants and symbols used see Table I. TABLE I Symbols and Constants x = c =
h = m, = mp = t
=
G = 7 =
N’ = N m, = Nm,=
H
=
He 0
=
Geometrical constant Velocity of light Planck’s constant Mass electron Mass proton Electronic charge Gravitation constant G. N. Lewis temperature unit = ~ / k Avogadro’s number Atomic weight of electron ” ” ” proton ’’ ” hydrogen ” ” ” helium I, oxygen j’
’j
,f
fi = Reduced mass =
3.14159 2.99796 6.547 8.994 1.6610 4.770 6.664
X X X X X X
cm sec-l 10-~’erg sec IO-^^ grams IO-?^ grams 10-l~absolute es units IO-^ cm3 gr-’ sec-?
IOIO
7.289 X iois 6.064 X 1oS3mole j.454 X IO-^ 1.0072346 1.00778 4.00216 16.000
8.9891 X ~ o - ~ ~ g r a m s mp+m0 mp/m, = Ratio mass proton to mass electron 1846.8 MEe = Loss of mass in formation of helium 4.776 X IO-^^ grams. 2i3.18OC To = Ice point, absolute 1.013250 X 106 dynes cm-? A,, = Normal atmosphere 22.4141 x 103 cms mole-’ V, = Gram molecular volume
J. E. MILLS
1090
TABLEI (Continued) R = Gas constant
=
A, V, -
8.3136 X
IO?erg
deg-I mole-'
T O
k cy
= =
Boltzmannconstant = R / N =
I/T
1.3710 X 1o-I6erg deg-'
2re2
0.7284 x
= -
Fine structure constant
/3 = Specific heat constant F = Faraday = Ne
hc h/k
=
4.7753 X 2.89253 X
v = Velocity of reduced mass in first Bohr orbit =
10-2
IO-'' 1014
sec deg es
2TE2
cms/sec h
a = Radius of first Bohr orbit w =
Period of reduced mass, first Bohr orbit
=
(y)' 5 (y2)'
=
2i (T)
=
v/c
=
Ek = Kinetic energy of reduced mass, first Bohr orbit
RH = Rydberg constant for hydrogen
sec
ergs
(Spectroscopic value of constants should v = Wave frequency be used) = Wave length of light in cms per sec. V = Wavenumber
2TE2
*
X
Atomic weights are from Aston. The values of the other constants are taken mainly from Birge: Phys. Rev. Sup. 1, I (1929).
As more and more functions were calculated it became evident that, ignoring the decimal point, nearly all of the functions calculated fell into a comparatively few series of figures. These series of functions and their reciprocals are given in Tables 11-XIV. TABLEI1 Functions related to c
1
c 1/12
(2) m.
e ; I
6
-
2 Ta C2
7pxvB 8 -
h2
2ne2
2.99796 X 3 . 0000
x
IO~O
Reciprocals
,33356 X ,33333
x
10-l~
10
.33339
3.0009 X
109
,33323 X IO-^
2.9981 X
10'
,33354 x
1
3.0000 X
103
'33333 x
I C 3
,33353 X
103j
2.9995
2.9982 X IO-^^
10-1
8
109I
RELATIONS BETWEEN FUNDAMENTAL PHYSICAL CONSTANTS
TABLE I1 (Continued) Functione related to c C2
9 c4z 2nh2
IO
11* G 8-
,9989 X
Reciprocals
1 0 ~ ~
,33346 X IO-^^
1og4
,33339
x
IO-^^
103
,33312
x
IO-'
10-l~
,33344 X d4
IO-'
,33366 X
103
,33370
IO-16
,33398 X
2.9995 X IO-^
.33339 X
103
3.0102 X 1oZ3
.33220
x
10-28
2
2.9995
x
3.0019 X 26
I2
13 I4
z/m,
2
2nt
2.9971
3Gc
2.9967 X
2 6.8 hc
37*
2
X 18i6.8 m,
2
,9990 X
,9942
x x
x
108
IO-'
TABLEI11 Functions related to c2 I
c2
2
22
Reciprocals
8.9878 X iozo 9.0000
,11126 X
9,0054 x
IO"
,11104 X
8.9886 X
1 0 ~ ~
,11125
x
10-16
IO'
,11111
x
10-6
9 .oooo
x
8.9892 X 8.9934 X I1
* 1846.8G2
12
~
,11124 X 1
0
x
IO'
8.9940 X
IO-''
8.9825 X
10-l'
9.0114
4e2
m,
~
~
1048
.I1097
x
IO-'
9.0093
x
10''
9.0139
x
10
,11094
,11119
x
d L 1846.8G
24*
1846.8h
8.9762 X loa
.1r14o X
n m 8 G
8.9969 X
.I1115
25
1070
x
x ,11133 x .I I I O 0 x
104
10-l~
.I1119
16*
€3
IO-'O
,11111
x
IO28
1018 10-10
IO-'
IO' 106
1092
J. E. MILLS
TABLE IV Functions related to c4
c4 2* 34
80.780 X 81.000
I*
8 -
e
81.064 X IO-^
hc2
12* m,c2 1846.8
IS*
Reciprocals
,12379 X ,12345 x
1040
€2
r6*
-
m,
46
e
10-1
,12336 X
105
80.836 X
IO-*
,12371 x
IO?
81.168 X
IO?O
.12320 X
IO-~I
x
IO?
,12307 X
IO-3
81.255
1846.8G h 21 moc2
10-ll
80.991 X IO-^*
.I2347
x
1021
81.024 X
.12342 X
10~’
10-l’
TABLEV Functions related to c6 I * c6
2
36
.7260 X
1 0 ~ ~
,7290 X
103
Reciprocals
1.3773 x 1,3717 X
IO+
IO-63
*
,7284 X 1o-I .;286 X
10-l~’~
,7283 X
10’
,7286 X IO-^
h
Io
,7284 X
IO-*
,7286 X
1017
1.3725 x
IO+‘
,7285 X
10-l~
1.3727 X
1015
1.3728 X
10~’
1,3733 x
IO?
,7284
,2
X
1
0
,7281 X
IO-’
,7285 X
10’~
*
.728j X IO-^ 1
I4
I7
v.5
. j288
X
.7280 X
~
~
~
1.3727
x
1.3723 X
IO*
x
IO2
IO-’
1.3721
105’*
1.3736 X IO-'^
1093
RELATIONS BETWEEN FUNDAMENTAL PHYSICAL CONSTANTS
TABLE V (Continued) Functions related to c6
3 I8 znh 19* 7 = I/k
*+ 20 21
h h/m,
* h3 (2 K t 2 )
22
x .7294 x ,7294 x ,7279 x ,7293
,7283 X
I026 10'6
10 IO 1 0 ~ ~
,7291 x 1o-Q/2
Reciprocals
x 1.3710 x 1,3709 x 1.3737 x 1,3731 x 1.3712
1,3715 x
IO-26
IO-16 IO-' 10-1
I0-O
IOQr2
35 m,% c%
TABLEVI Functions related to c8
C
Reciprocals
6.5254 X IO^ 6.5610 X 103 6.5463 X 1 0 ' ~
,15324 x ,15242 X ,15276 X
6.5480 X 1oS7
,15272 X
6.5476 X 105 6.547 X IO-?' 6.5467 X 1o15
,15273 X IO-^ .15274 X 1oZ7 ,15275 X IO+
6.5497 X
,15268 X 1oZ7
IO-?'
6.5557
IO-*
10-8 10-l~
10-8'
. '5254
TABLEVI1 Functions related to c*o I
C'O
2
3'0 hc2
IO
5.8649 X 10104 5.9049 X 104 5.8843 X IO-^
Reciprocals
,17050 X 1 0 - l ~ ~ ,16935 X IO-^ ,16994 X IO(
This series could be expanded, but the functions do not seem to be of importance.
J. E. MILLS
I094
TABZEVI11 Functions related to el2
x ,53144 x
.53052 X
IO-'
,53088 X
10-l'
1.8837 X IO'^
IO-4
I
53051
c 'I*
a m 8 G
Reciprocals
1.8971 X IO-@ 1.8817 X IO-^ I 8850 X I O
,52712
10126 IO'
x
I
,8850 X
104
.53086 X 1oS4
1.8837 X IO-^^
,53072 X I O - ~ O ,53061 X IO-^^
I ,8842 X IO*O 1.8846 X 1og4
, 5 3 0 1 ~X IO-^
1.8861 X
$3070 X 53103
x
X
104
1013
I . 8843
IO'
1.8831 X IO-^
10-l~
,53110 X IO-^
I
,8829 X
,53000~X IO^
I
,8868 X IO-^
,53161 X IO-^
1.8811X
1oY
52768 X 1ol1
1.8951 X
10-1'
104
TABLEIX Functions related to elr
Reciprocals , 2 1 1 0 7X 1 0 - l ~ ~
4.7377 X 4.7830 X
1
106
,20975 X
4.7714 X
109
,20958 X IO-^
0
~
~
~
4.7667 X 1oa8
,20979 X
4.7681 X iois
.20973
4 . 7 706 X iob4
,20962 X
x
IO&
10-16
IO-^(
109j
RELATIONE BETWEEN FUNDAMENTAL PHYSICAL CONSTASTS
TABLE IX (Continued) Functions related to
IO
11* I2
h2
c2 tC?
7rdl846.8G C2
-
2TdK C2
I3
Fe
x
Reciprocals
el4
,20964 X
IOIO
4.7690 X IO-'^
,20969 X
1
4.7651 X
1 0 ~ ~
,20986 X
10-l~
4.7698 X
1 0 ~ ~
4.770
9 e
IO-''
0
~
~
20965 X IO-^^
4.7728 X 1ol2
C
,20952 X
IO-'?
x
IO-16
,20950
I4 3rG 4.7780 X IO'^ 26*
TG 47r (1846.8)~
27
4,7765 X
106
,20936 X
IO+
4.7685 X
10-j~
,20971 X
1014
4.7718 X 4.7757 x
1
,20956 X ,20939 x
1 0 ~ ~
4.7684 X
1014
,20971 x
4.7653 X
1 0 ~ ~
,20985 X IO-^^
Io-'1
,20941 x
C2
28
p a
33*
mHe
EC
35
zo
3' 47*
4n(1846.8)~ m, P = h/k
4,7753
x
0
~
~
IOz6 IO-14
IO"
TABLEX Reciprocals
Functions related to elb
42.581 43.047 42.884 42.872 42.863
X
1
x
IO6
0
X IO*^ X IOIO X IO-^(
42 ,931 x
IOz6
42.859 X 106 42.974 42.861 X IO-^^ 42.923 x IO-6
~
~
~
023483 X 10-l~~ ,023231 X IO-^ ,023319 X IO-?^ ,023325 x 10-10 ,023330 X 1 0 ~ ~ ,023293
x x
,023332 ,023270 ,023331 x ,023joo x
10-26 10-6
10'' I06
1096
J. E. MILLS
TABLEXI Functions related to c L 9 I* c18 2* 315
31*
34* 39
v'I846.8
c?
-&
3.8271X 3.8742x 3.8624X 3.8737x
1846.8G
i~
Reciprocals
1oZ2
,26127X ~ o ,25812X IO-^ ,25890X IO+
103
,25815X IO-^
1oIa8
I08
3.8664X IO-^
,26864X
-
~
~
~
104
TABLEXI1 Functions related to ca* I * c5? 2* 352
IO*
h4
32* (PV2)*
40 mp/mo
Reciprocals
1813.1X 1033~ 1853.0 X 1o12 1837.25X I O - ~ O ~ 1837.1X IO-^^ 1846.8
,55150 X ,53966X 10-l~ ,54429X ioxo5 ,54434x Ios1 ,54148x 10-8
1847.8 x
,54118 X
IO-$
106
TABLEXI11 Functions related to c34
Reciprocals
I* c34
I .6296X 103j6
2* 354
1.6677X 1ox6 1.6610X 10-*~
.61360X .59962X ,60205X
10-l~
1.6656X IO-^
,60038X
104
1.6647X
,60072X
IOO
42* mp e
43 44
dI846.8G
v F
1846.8
io5
1
0
1oZ4
TABLEXIV Functions related to cS I 2
34
*
cas 336
;
42* mp c2 45 3/2
Reciprocals
,14646X ,15009x
IOx8
6.8278X 6.6625X IO-^^
.15006X
108
6.664 X IO-^
,14929x
IO-?
6.6985X 6.6667
. I5000
10~'~
IO?
~
~
~
~
RELATIONS BETWEEN FUNDAMEKTAL PHYSICAL CONSTANTS
IO97
I098
J. E. MILLS
I n these tables closely related functions are given the same number. I n each table one of these numbers is italicized and all functions different numerically from the value of this italicized function by more than one part in a thousand are marked with a n asterisk above the number. Again ignoring the decimal point (see section 6) it will be a t once evident that the agreement shown between the figures given in each table could only occasionally be accidental. It was finally found that all of the different series of figures are related one to another and that apparently all were related to the velocity of light. This relationship is brought out clearly in Table XV. In columns I and z are shown even powers of 3 to 336. Similarly in columns 7 and 8 are shown even powers of the velocity of light to c36. Ignoring the decimal point the velocity of light differs from 3 by about one part in fifteen hundred. Nevertheless when both have been raised to the thirty-sixth power the divergence is increased to more than two per cent. The basic constants and a few derived constants are shown in columns 3 , 4, 5 , and 6. All of these constants (except x ) and all of the derived functions shown, and nearly all in the extensive series of functions of which they serve only as examples, lie in value between the respective powers of 3 and c. This result is amazing. I t follows a t once that all other similar functions involving these basic constants will similarly be related numerically through the velocity of light. One may ask “How can light cause the mass of the electron, the electronic charge, Planck’s constant and other constants to have their present values?” A little reflection will show that one has an equal right to turn this question backwards and ask “How is it that the mass of the electron, the electronic charge, Planck’s constant and other constants, cause light to have its present velocity?” Still further reflection will show that Table S V really indzcates simply a numerical relationship between these basic constants and lzght and that a similar table could be constructed using a n y one of the constants as a basis. From this standpoint there is nothing necessarily unique about the velocity of light. The importance of Table XV as it stands lies in the fact that it ’ reveals a hitherto unknown numerical relationship between the fundamental constants shown. It becomes important therefore to study this numerical relationship in detail. 1. Could the Relationship shown be Accidental? U’hen the first few numerical agreements between the gravitational constant and other constants were noted when dealing with functions, which were not known to be related and which differed in their assigned dimensions, the author was skeptical as to the valuable character of such agreement. As the number of similar functions increased, common sense and experience with calculations were sufficient to enable one to conclude with certainty that the relations were not merely accidental coincidences. Now it is quite possible to state the relations in such form that anyone who desires can apply the laws of probability.
RELATIONS B E T W E E N FUNDAMENTAL PHYSICAL CONSTANTS
1099
Ten numerical functions are involved as follows: 2,
31
x , C, m~ Or I*, k, h, E, m ~ G,a
The mass loss in the formation of helium is omitted from the list as no one is certain of the derivation of the cosmic rays. I t may be granted, for the present, that the similarity between 3 and 2.99796 is accidental. Moreover G. N. Lewis has shown that h can be calculated from function 30 of Table VI. This reduces the supposedly unconnected numerical constants to eight. Moreover G. N. Lewis has recognized the fact that ordinary equations connect'ing some of the remaining constants can be used to derive what he calls ultimate rational units, and has stated that properly defined units will produce always simple numbers. H e has made further use of this fact to calculate h, the constant of Stefan's law and a constant in the equation for the entropy of monatomic' gases. His reasoning has been rather vigorously disputed.2 Bridgman states that Planck was the first to suggest a system of ultimate rational units. These articles do show that extensive numerical simplification would result from changing our system of units. But the articles do not prove the complete numerical simplicity that exists in our present system (see Table XVII) nor show the connection between this simplicity and the velocity of light. Nor do they prove numerically simple relations between the masses of the proton and electron and gravitation. Moreover the views, as advanced, have not been sufficiently proved to lead to a general recognition of their validity. Now one might assume roughly that each constant is measured to an accuracy of one part in one thousand. In calculating any given function from a combination of these constants, if there is no actual relation between the constants, a new numerical series should be introduced not only with each combination of constants but with every variation in the power of any constant. An inspection of the functions calculated will show that the constants are used in some of the various functions to the variation in the powers shown below : Functions 2 x c m,orp k h E mD G Variation in power used 4 5 4 4 I 3 5 5 3 Both the laws of probability and the theories of numbers will show that combinations of unrelated constants as made could produce agreeing numerals only rarely. (Functions that contain 3 as a factor may be based on wholly accidental agreement). Any explanation of the numerical agreement shown as being a mere series of coincidences is utterly impossible. Still more imposG. N. Lewis and E. Q. Adams: Phys. Rev., (2)3, 92 (1914);G.N. Lewis: Phil. Mag. 45, 266 (1923);49, 739 (1925); G. N. Lewis, G. E. Gibson and W. M. Latimer: J. Am. Chem. SOC.,44, 1008 (1922). *Norman Campbell: Phil. Mag., 47, 159 (1924);0.J. Lodge: 45, 276 (1923);49, 751
(1925).
I100
J. E. MILLS
sible would it be for the various series of numbers showing such agreement to be accidentally related to each other and to the velocity of light in the manner shown.
The numerical agreements shown must be recognized as a basic fact connecting physical constants. Yet two more facts must be recognized as having a direct bearing upon any possible accidental numerical agreement among the various functions. a) Strange as it may seem from the character of the functions calculated, very few of these functions were originally calculated for the purpose of showing numerical agreement. Almost all of the forty-seven functions, shown in Table XVIII, to which the others can be immediately reduced, were either previously recognized as being of “natural” significance, or were calculated by the author in the progress of certain theoretical investigations. b) As shown under Section 5 , headed “Can Exact Numerical Agreement be obtained?’’ only slight changes are necessary in the values of the constants a t present adopted to make nearly all of the functions shown in any given series give numbers agreeing to one part in five thousand. With very many functions there would be perfect agreement. 2. The Exactness of the Agreement
Of all of the results shown in Tables 11-XIV only the following differ from the underscored function number in each table by more than one part in five hundred (ignoring the decimal point). Powers of c and 3. c and 3 differ as to the numerals involved by only about one part in 1500. Nevertheless when the numbers are raised to high powers, a serious multiplication of this difference is of course evident. Functions 37 Table 11, 1 5 Table IV, 16 Tables I11 and IV, 36 Table VIII, and 3 1 Table X, all contain 1846.8as a factor. As is well known, calculations based on spectroscopic evidence lead to the value of this ratio as about 1838. Calculations based on deflection measurements lead to the value 1846.8. Apparently some such divergence is at the bottom of the disagreement shown by these functions. The recent discovery of a n isotope of hydrogen must also be considered. Functions I O and 32 of Table XI1 give 1837.25 and 1837.1 as numerals, confirming evidence based on spectroscopic determinations. Function 42 of Table XIV is the only function included in the tables differing from the expected numeral by more than one part in 500 for whose divergence we can a t present find no reason. A few other additional functions as shown by an asterisk differ from the underscored function by as much as one part in a thousand. These differences usually do not probably arise entirely from the uncertainty of the basic constants used. They arise from the fact that 3, m,, and p are probably not always correctly used and from the further fact that certain existing disturbances (as for instance a disturbance caused by the physical dimensions of the particle) should be allowed for and were not. It seemed best to the
RELATIONS BETWEEN FUNDAMENTAL PHYSICAL CONSTANTS
I IO1
author not to attempt to apply minor corrections to any function, since such a n attempt might lead to the feeling that the agreements had been obtained by unjustifiable modifications. 3. The Dimensions of the Functions compared
The author has already been accused even before the publication of this paper of paying no heed to the dimensions of the functions compared. This may indeed be a grievous error. But if it is, the author is not to blame. The mistake was made by nature. All the author has done is to show that a numerical relation actually exists between quantities of different dimensions. Nature, not the author, must be called on to explain why. The author indeed has no intention of engaging in the oft repeated discussion of dimensions. I t is commonly granted that magnetic permeability and the dielectric constant are mutually related to the velocity of light. That discovery marked a big step forward and leaves us with only three dimensions concerning which to worry-mass, length, and time, since temperature is known to be related to kinetic energy. The relativity theory has insisted upon a connection between length and time and the velocity of light is concerned also with that relation. The implied connection found by the author is of a very different character, but again concern3 the velocity of light and at any rate the imagination is not further strained. This leaves only mass to worry about. No one knows what causes mass, but it is a step forward to find that mass also is connected with light and time, and again through the velocity of light. Others have advanced ideas concerning the possible electromagnetic nature of mass. Such theories presuppose the possibility of omitting mass as a n ultimate dimension. The idea of dimensions can be made to serve a very useful purpose, but there is often times confusion in their use. One is quite accustomed to the idea that m v represents the momentum of a moving particle and I/Z mv? its kinetic energy. But the fact that we have chosen to square the velocity of the particle, and give the name energy to the function thus obtained, has not changed the actual velocity of the particle one iota. The particle still has the same velocity and the dimensions attributable to the particle are still ml/t. We have merely chosen to take two other dimensions l/t, which really belonged to the space traversed by the particle, and have attributed these dimensions also to the particle. This is mathematically very convenient, but it should not be allowed to obscure the fact that energyis in reality a function of the space traversed as well as of the particle, and is dependent on both. So mass should be regarded as a dimension of a particle only until we realize the true nature of the ultimate property which is its cause. Personally the author believes ultimately that mass will be found to be a property of the physical dimensions of the particle concerned and of the energy of the surrounding space. I n other words, it is a property concerned with the distribution of energy in space.
J. E. MILLS
I I02
4. The Relationship between 2, 3, x , and c
At first one of the most baffling phases of the numerical relations found to exist was the fact that ignoring the decimal point the following relations were very approximately true:-
TABLEXVI Relations between 2, 3, 2
3 3/2
n 2 8
Corresponding Value of c 3 .00005 3 .ooooo 2.99995 3.00019 2.999601
= r/c= = c = c36 = =
(322
I/C’3
x
and c Corresponding Value of c
All of the functions can be derived from those for 2 8 , 4 n and 6 x 1 or from any three of the functions that contain the factors 2 , 3 and x . The equations do not necessarily exist simultaneously in nature with regard to the same functions and the author thought that a careful determination of the corresponding value of c might lead to the rejection of some one of the relations. But the greatest divergence of any of the values of c from the mean value is as shown inTable XVI, only one part in IO,OOO. Another surprising result of this check is that all values of c are considerably above the actual observed value. For a discussion of this point see under Section 5 . Actually in nature the relationship probably arises from such equations as I/ZTC
=
a function which reduces numerically to cl*, or
4 n ( 1 8 4 6 . 8 ) ~ = a function which reduces numerically to cI6
and from the possible accidental relation numerically of 3 to 2.99796. If the natural relations concerned could be determined all of the others could be discarded as of no consequence in spite of their very close similarity numerically to other naturally arising functions. Curiously enough the first Bohr orbit of the electron and proton is almost exactly IOOO/C and the orbit which would correspond with a loss of energy equivalent to Millikan’s longest cosmic ray is I / I O O O C . The corresponding radii of the orbits are I O O O / ~ T C and I / Z O O O ~ C . Other unique points in nature give rise to similar simple relations and serve t o make clear that such relations concerning a as those shown above will exist. Why the orbits are a t the points stated is a much more baffling problem. No reason is known why 8 Q G should be related to c7,* nor why 3n should be related to cl* X IO-^'^. 5. Can Exact Numerical Agreement be obtained? The best way to answer this question is to choose a single function and calculate all values and constants from this function. For this purpose, we * See G. N. Lewis and E. Q Adams Phys. Rev. ( 2 ) 3, p 92 (1914);
I 103
RELATIONS BETWEEN FUNDAMENTAL PHYSICAL CONSTANTS
choose the function 2 x c = I/@. The choice of the function may be regarded for the present as somewhat arbitrary although it was selected because the radius of the first Bohr orbit as stated above is I O O O / ~ T C . The results of the calculation are given in Table XVII.
TABLE XVII Calculation of Physical Constants from the Equation Observed C
C
C2
m, 1/ k h e mP 1/ G
C6 C8 C14
c34 CS6
1/2m
= cI2 Calculated
(2.99796 ~0.00004) X IO‘’’ (8.994 + 0.014) X IO-^^ (7.294 f 0.0074) x IO15 (6.547 0.008) X 10-*~ (4.770 zk 0.005) x IO-’’ (1.6610 I 0.0017) x IO-z4 (1.5006 i 0.0005) X IO?
*
2.99960. 8.99761 7.28415 6.55402 4.77401 1.66014 1.49373
The decimal points have again been ignored in making this calculation (See Section 6). The divergence of I / k is probably due to the size of the molecules of the gas. This point will be considered in a later paper. As regards the velocity of light it should be borne in mind that if an ether does exist made up of discrete particles, and if a mechanical explanation of the universe is possible, then the velocity of light will differ from the velocity of the particles that cause the wave. Table XVII a t the present time should be taken only as a n indication of the possible aid that may be obtained from the relations shown in determining more accurately the value of physical constants. The author regards the constants as calculated a t the present time as possible “ideal” constants that would be produced if the ether particles were reduced to points which retained their present properties. Use can be made of the facts brought out by the table to bring into prominence certain simple relations which should aid in arriving a t the underlying cause of the relations shown. Thus it is useful to recognize that using the calculated constants Gmomp = IO-^^ exactly, and that numerous similar exact relations will hold. The facts given may furnish a clue to the cause of the present discrepancy between certain spectroscopic and deflection measurements. There are numerous other possibilities suggested. The author was led by a simple speculation in connection with the facts to the discovery that the mass loss causing Millikan’s longest cosmic rays is numerically 7.294 times Planck’s constant and Planck’s constant is of course numerically 7.279 times the mass of the electron. One may indeed be allowed to guess that perhaps the mass loss represented by these rays is more accurately 7.284 times Planck’s constant, and that the mass loss represented by the shortest cosmic rays is 7.284 times the mass loss represented by the longest rays.
J. E. MILLS
1104
The first Bohr orbit of the electron and proton is rooo/c in length and the orbit under the same law required for Millikan’s longest cosmic rays is I/IOOOC. Is it possible that these cosmic rays have their origin in the formation of the proton itself? Is it possible that the proton is produced by two electrons moving in a n orbit with semi major axis 1.526X 1 0 - l ~centimeters? Is it possible for a n electron to move in an orbit which would normally give to the electron a velocity above the velocity of light? Does the motion of an electron in such an orbit give rise to a positive charge and a large increase in mass? Is the electron an ether particle which has “lost” some of its velocity? At the present time these are mere speculations caused by an attempt to follow back ideas suggested by the relations shown. They are cited here only to show that the facts followed back suggest possibilities from which perhaps the false may be sifted. 6. The Decimal Point
I n calling attention to the numerical relationships the decimal point has been ignored. I n considering the entire function the decimal point cannot be ignored. As explained, Table XV is artificial in that it appears to make all of the constants depend numerically on the velocity of light. What the table really indicates is a numerical relation among the various constants shown and any constant shown could be made the basis for the table. The correct ultimate table would probably (from the author’s point of view) be based in part on more fundamental constants. Between certain constants reciprocal relations exist and one or more physical entities which partly give rise to a property are eliminated. Thus time does not appear directly in Table XV nor does the volume of the masses. The constants really concerned in the relations appear to be 2, 3, T and c. Using some of these numbers above it is quite possible to write such functions 3 l1 3 3 8 I - 337 - - -, where the 3 is more exactly 2.9996. Numerous as h = -, m p = c3 c4’G c similar relations could be written all of no value unless the true relations existing in nature were found. Introducing some theoretical considerations the author suspects strongly that 2, 3, H , and c will all be concerned in the equations which really represent nature. Thus G appears numerically to be related to ca6. In reality the relations G = 2ooo/c and I / T G = 4.7765 X 106 are probably a somewhat nearer approach to nature, nor is the possible introduction of a logarithmic relation absurd, either theoretically or numerically. It should be realized that the ratio of the functions in any one of the Tables 11-XIV will give a power of I O very nearly, and that if the calculated constants shown in Table XVII are used then the ratio will often be a n exact power of IO. When this fact is considered and it is further realized that we must go one step back of our present physical constants in representing nature in the simplest manner then the fact that we have ignored the decimal point in deriving the relations existing with our present units should not cause any feeling of skepticism. There is nothing mystical concerning the relations.
-
RELATIONS BETWEEN FUNDAMENTAL PHYSICAL CONSTANTS
I105
7. Summary of Functions calculated A summary of the more important functions calculated are shown in Table XVIII. It appears somewhat strange that the natural functions seldom lead to simple numerical relations involving odd powers of c, though of course a few such functions have assumed some importance.
I C
TABLE XVIII Summary of Functions 2.99796 X I O ~ O
2 3
3.00000
13
2
2.9971 x 2.9967 X
m
3Gc I4 37 12 25
IO-'
52.768
X
109
53.054
x
IO-'
53.000
X
1051~
4.770
x
IO-''
4.7757
x
10-26
103
I
3.0102 X X 1846.8m0 m, 8.994 X 7 r m G 8.9969 X
10~3
2
IO-**
IO&
4.7765 X
1o6
4.7681 X
10'~
€8
24 I5
1846.8h d-&ES 7 I
8.9762 X IO-^ 9.0093
x
IO"
16 ___ 18~6.8G
8.1255 X
h/m, 20 m d h 8 e/h 19 r = ~ / k
7.279 7.294 7.2860 X 1o16 7.2886 X 1015 7.293 x 1025
x
IO9
4.7746 X
IO-l
4.7714
108
4.7753 21
I8 3/2ah 22
IO
(2R€2)2 v2 -h3 h h
29 w = ZRHC C
7.283
X
6.547 6.5467
x x
6.5497 X
104?
28 pa=m,a,=m,a, 4~(1846.8)? 3' m, 27 4 ~ 1 8 4 6 . 8 ~ 32
10-27 1016
IO-*'
P>
31 V 1 8 4 6 . 8 39 n1846.8G mP 40
m,
41
d f i
42
m,
x
10-11
4.7718 X IO-s6 4.7653 X
1os4
42.859 x 1 0 8 42.861 X 1 0 - l ~ 42.9744 3.8664 X IO-^ 1846.8 1847.8
X
10-9
1.6610 X IO-?^ 0
43
1.6656 X
IO-'
1.6647 X
105
34 I/G
.15006 X
45 312
.I 5 0 0 0
108
I 106
J. E. MILLS
The functions shown in Table XVIII are all dependent upon the relations already discussed between the more fundamental constants 2 , 3, T,c, ma, k, h, e, mp, and G. The derived functions give however sometimes a n unexpected hint as to the nature of the relations between the fundamental constants. Actually a study of the derived functions led to the finding of the numerical relationship between the fundamental constants. Their further study may suggest the complete relationship. 8. Some Suggestions as to the Ultimate Meaning of the Facts
The author does not believe that there is anything in the facts brought out which justifies representing mass or time as a length, or which requires very radical readjustment of primary ideas. A simple numerical relation has been found to exist between fundamental physical constants and these simple relations would be extended by new definitions of atomic weights and certain other constants. But the physical realities existing remain as before. On the other hand the simple numerical relations proved to exist indicate actual relations between the physical entities concerned of a nature not yet fully realized. For example no simple numerical relation will exist between the mass of an atom of tin or lead and the velocity of light. If therefore a simple relation is found between the mass of the proton and the mass of the electron and the velocity of light a fact has been discovered which requires some sort of an explanation. For there is no a priori reason why the complexity of the proton and electron might not prohibit any simple numerical relation between them and between other physical constants. The author believes that the facts shown are of such a nature as to point rather strongly to the idea that a mechanical explanation of the universe is possible. Certain particles (masses) existed in the universe. A certain amount of motion (energy) was available, and finally distributed itself among the existing particles. As a result we finally have “created” the masses of proton and electron, electronic charge, Planck’s constant, the velocity of light, gravitation, and the various attractive forces and results dependent upon them. Protons and electrons apparently exist in equal numbers and the suspicion that both are “created” from some possible simpler particle, such as a n ether particle, is increased by the numerical relationships discovered. Following out the ideas above in a little more detail it seems to the author probable that an ultimate unit of time is desirable and probably would be best defined as the time required for an ether particle to traverse the space which it occupies due to its motion. This is in order to prevent undue multiplication of the space occupied by the particle in relations concerning its motion. Then if it be assumed that this fundamental space traversed in a unit of time bears a simple relation to the orbit which defines the proton and that the diameters of the particles themselves are not greater than 1/12 of the diameter of this orbit, the simple numerical relations shown might possibly follow.
RELATIONS BETWEEN FUNDAMENTAL PHYSICAL CONSTANTS
I 107
s-arg Simple numerical relations have been shown to exist between the constants R, m,, I/k, h, e, mp, and r/G, and the velocity of light, and therefore such relations exist between all functions calculated from them. I.
2. The fact that such a relation exists proves a connection between the constants not as yet understood.
3. The above constants can be calculated from the equation ~ / z m= cl* and their relation to certain powers of the velocity of light, if the decimal point is ignored. This indicates that the relation which exist,s is of a very simple nature. University of South Carolina, Columbia, S. C. January 21, 1932.
