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  1. “Graphical Exposition of the Michelson-Morley Experiment,” J. Opt. Soc. Am. 27, 177 (1937).
  2. This was recognized by Lorentz, who describes the corresponding factor in his expressions as“the origin of all our difficulties,” (Theory of Electrons, p. 219), and chooses the value corresponding to our n= 0 by consideration of the Kaufmann and Bucherer experiments. Birkhoff (Relativity and Modern Physics, p. 34) concludes that “the decision … must be made solely on the basis of facts established by means of physical experiments,” and (p. 55) “It appears that no experimental data are available.” He then discusses an experiment (the “transverse Doppler effect”) which would be decisive.
  3. Larmor, Aether and Matter, 1900.The characterization of the frequency change as a prediction of the theory of relativity (1905) is chronologically unjustifiable.
  4. If the observer is called upon to insure that the clock is moving with uniform velocity with respect to the body (which is moving uniformly by assumption) he must watch the passage of uniformly spaced divisions past the clock. These divisions can be those on our standard measuring rod whose uniform velocity with respect to the body (which might be zero) can be assured by independent observation, with the clock held stationary on the body. This procedure avoids any commitment as to the methods of length measurement on the moving body.
  5. See Appendix. I am indebted to Mr. T. C. Fry for the formal solutions.
  6. In the procedure used to obtain the Lorentz transformations by starting with the postulate that the velocity of light is always c, is the concealed restriction that all clocks and rods are to be manipulated at infinitesimal speeds. The final results are subject to this same restriction, which when recognized leads to such dicta as “clocks must be excluded from time measurement,” and that the times recorded by a clock are “incorrect.” In view of the role played in both macroscopic and atomic physics (e.g., spectroscopy) by rods and clocks, no system which throttles them can be physically satisfactory.
  7. It is possible, by assuming I and III, to derive II, by imagining light signals, sent over a to and fro path, to take an invariant measured time, and by requiring that the measured distance and time at the far point shall be invariant whatever the velocities of the rods and clocks used in measurement. Both this line of attack (assumption of invariance), and that followed in the paper (assumption as to behavior of material rods and clocks) requires I, which is thus characterized as the fundamental assumption in the treatment of light signals.
  8. The relativity treatments of stellar aberration all start with a“wave front” from a star, thus of necessity recognizing the pattern of radiant energy here considered as a manifestation of the ether. The common relativity treatment of aberration concerns itself with a broad section of a wave front, and shows that, due to the “local” time difference at points along the receiving surface the wave front will always be interpreted as being normal to the direction of propagation of the wave front section as a whole, with respect to the earth. This does not touch the question of the cause of the direction taken by the wave section or ray, and is incidentally not the case of a star image formed by a telescope. The second-order term in the “relativity” expression for aberration follows at once along the lines here followed, from the contraction of the measuring plane in the moving telescope.

1937 (1)

1900 (1)

Larmor, Aether and Matter, 1900.The characterization of the frequency change as a prediction of the theory of relativity (1905) is chronologically unjustifiable.

Larmor,

Larmor, Aether and Matter, 1900.The characterization of the frequency change as a prediction of the theory of relativity (1905) is chronologically unjustifiable.

Lorentz,

This was recognized by Lorentz, who describes the corresponding factor in his expressions as“the origin of all our difficulties,” (Theory of Electrons, p. 219), and chooses the value corresponding to our n= 0 by consideration of the Kaufmann and Bucherer experiments. Birkhoff (Relativity and Modern Physics, p. 34) concludes that “the decision … must be made solely on the basis of facts established by means of physical experiments,” and (p. 55) “It appears that no experimental data are available.” He then discusses an experiment (the “transverse Doppler effect”) which would be decisive.

Aether and Matter (1)

Larmor, Aether and Matter, 1900.The characterization of the frequency change as a prediction of the theory of relativity (1905) is chronologically unjustifiable.

J. Opt. Soc. Am. (1)

Other (6)

This was recognized by Lorentz, who describes the corresponding factor in his expressions as“the origin of all our difficulties,” (Theory of Electrons, p. 219), and chooses the value corresponding to our n= 0 by consideration of the Kaufmann and Bucherer experiments. Birkhoff (Relativity and Modern Physics, p. 34) concludes that “the decision … must be made solely on the basis of facts established by means of physical experiments,” and (p. 55) “It appears that no experimental data are available.” He then discusses an experiment (the “transverse Doppler effect”) which would be decisive.

If the observer is called upon to insure that the clock is moving with uniform velocity with respect to the body (which is moving uniformly by assumption) he must watch the passage of uniformly spaced divisions past the clock. These divisions can be those on our standard measuring rod whose uniform velocity with respect to the body (which might be zero) can be assured by independent observation, with the clock held stationary on the body. This procedure avoids any commitment as to the methods of length measurement on the moving body.

See Appendix. I am indebted to Mr. T. C. Fry for the formal solutions.

In the procedure used to obtain the Lorentz transformations by starting with the postulate that the velocity of light is always c, is the concealed restriction that all clocks and rods are to be manipulated at infinitesimal speeds. The final results are subject to this same restriction, which when recognized leads to such dicta as “clocks must be excluded from time measurement,” and that the times recorded by a clock are “incorrect.” In view of the role played in both macroscopic and atomic physics (e.g., spectroscopy) by rods and clocks, no system which throttles them can be physically satisfactory.

It is possible, by assuming I and III, to derive II, by imagining light signals, sent over a to and fro path, to take an invariant measured time, and by requiring that the measured distance and time at the far point shall be invariant whatever the velocities of the rods and clocks used in measurement. Both this line of attack (assumption of invariance), and that followed in the paper (assumption as to behavior of material rods and clocks) requires I, which is thus characterized as the fundamental assumption in the treatment of light signals.

The relativity treatments of stellar aberration all start with a“wave front” from a star, thus of necessity recognizing the pattern of radiant energy here considered as a manifestation of the ether. The common relativity treatment of aberration concerns itself with a broad section of a wave front, and shows that, due to the “local” time difference at points along the receiving surface the wave front will always be interpreted as being normal to the direction of propagation of the wave front section as a whole, with respect to the earth. This does not touch the question of the cause of the direction taken by the wave section or ray, and is incidentally not the case of a star image formed by a telescope. The second-order term in the “relativity” expression for aberration follows at once along the lines here followed, from the contraction of the measuring plane in the moving telescope.

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[ ( 1 υ 2 / c 2 ) 1 2 ] n + 1 : 1
[ ( 1 υ 2 / c 2 ) 1 2 ] n : 1
[ ( 1 υ 2 / c 2 ) 1 2 ] 1 n : 1 ,
( υ + W ) T = L s ( 1 υ 2 / c 2 ) 1 2 + υ T ,
T = L s ( 1 υ 2 / c 2 ) 1 2 / W
T b = T [ 1 ( υ + W ) 2 c 2 ] 1 2 = L s ( 1 υ 2 / c 2 ) 1 2 W [ 1 ( υ + W ) 2 c 2 ] 1 2 .
T a = T ( 1 υ 2 c 2 ) 1 2 = L s ( 1 υ 2 / c 2 ) 1 2 W ( 1 υ 2 c 2 ) 1 2
Δ S = L s ( 1 υ 2 / c 2 ) 1 2 W [ ( 1 υ 2 c 2 ) 1 2 ( 1 ( υ + W ) 2 c 2 ) 1 2 ] ,
Δ T = L s / W [ ( 1 υ 2 / c 2 ) 1 2 ( 1 ( υ + W ) 2 / c 2 ) 1 2 ] .
L i = L s ( 1 υ 2 / c 2 ) 1 2 Y Δ T
L i = L s [ ( 1 υ 2 / c 2 ) 1 2 ( Y / W ) { ( 1 υ 2 / c 2 ) 1 2 ( 1 ( υ + W ) 2 / c 2 ) 1 2 } ] ,
L i = L s [ ( 1 υ 2 c 2 ) 1 2 Y W { ( 1 υ 2 c 2 ) 1 2 ( 1 ( υ + W ) 2 c 2 ) 1 2 } ] ( 1 ( υ + Y ) 2 c 2 ) 1 2 .
Y 0 = Y ( 1 υ 2 / c 2 ) 1 2 ( 1 ( υ + Y ) 2 / c 2 ) 1 2 ,
from which Y = Y 0 ( 1 υ 2 / c 2 ) Y 0 υ / c 2 + ( 1 + Y 0 2 / c 2 ) 1 2 .
W 0 = W ( 1 υ 2 / c 2 ) 1 2 ( 1 ( υ + W ) 2 / c 2 ) 1 2 ,
from which W = W 0 ( 1 υ 2 / c 2 ) W 0 υ / c 2 + ( 1 + W 0 2 / c 2 ) 1 2 .
L i = L s ( 1 υ 2 / c 2 ) 1 2 Y 0 ( 1 υ 2 / c 2 ) Y 0 υ / c 2 + ( 1 + Y 0 2 / c 2 ) 1 2 W 0 ( 1 υ 2 / c 2 ) W 0 υ / c 2 + ( 1 + W 0 2 / c 2 ) 1 2 [ ( 1 υ 2 / c 2 ) 1 2 { 1 ( υ + W 0 ( 1 υ 2 / c 2 ) W 0 υ / c 2 + ( 1 + W 0 2 / c 2 ) 1 2 ) 2 c 2 } 1 2 ] [ 1 + ( υ + Y 0 ( 1 υ 2 / c 2 ) Y 0 υ / c 2 + ( 1 + Y 0 2 / c 2 ) 1 2 c 2 ) 2 ] 1 2 .
L i = L s [ ( 1 + Y 0 2 / c 2 ) 1 2 + ( Y 0 / W 0 ) ( 1 ( 1 + W 0 2 / c 2 ) 1 2 ) ] .
L i = L s [ ( 1 + Y 0 2 / c 2 ) 1 2 + Y 0 W 0 / 2 c 2 ] ,
L i = L s .
T t = L s ( 1 υ 2 / c 2 ) 1 2 c υ ,
T i = L s ( 1 υ 2 / c 2 ) 1 2 ( 1 υ 2 / c 2 ) 1 2 c υ = ( L s / c ) ( 1 + υ / c ) .
T t = L s c ( 1 + υ c ) L s ( 1 υ 2 / c 2 ) 1 2 W 0 ( 1 υ 2 / c 2 ) W 0 υ / c 2 + ( 1 + W 0 2 / c 2 ) 1 2 [ ( 1 υ 2 c 2 ) 1 2 ( 1 ( υ + W 0 ( 1 υ 2 / c 2 ) W 0 υ / c 2 + ( 1 + W 0 2 / c 2 ) 1 2 ) 2 c 2 ) 1 2 ] .
T t = L s c [ 1 + W 0 / c ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ] .
T t = ( L s / c ) ( 1 W 0 / 2 c ) ,
L i T t = L s L s / c [ ( 1 + Y 0 2 / c 2 ) 1 2 + Y 0 / W 0 ( 1 ( 1 + W 0 2 / c 2 ) 1 2 ) ] [ 1 + W 0 / c ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ]
= c L / T = c m ,
t = d / c .
d 1 = d ( 1 υ 1 / c ) , d 2 = d ( 1 υ 2 / c ) , d n = d ( 1 υ n / c ) .
d ( 1 υ 1 / d ) ( 1 υ 1 2 / c 2 ) 1 2 .
d 1 = d ( 1 υ 1 / c ) ( 1 υ 1 2 / c 2 ) 1 2 [ ( 1 + Y 0 2 / c 2 ) 1 2 + ( Y 0 / W 0 ) { 1 ( 1 + W 0 2 / c 2 ) 1 2 } ] , d 2 = d ( 1 υ 2 / c ) ( 1 υ 2 2 / c 2 ) 1 2 [ ( 1 + Y 0 2 / c 2 ) 1 2 + ( Y 0 / W 0 ) { 1 ( 1 + W 0 2 / c 2 ) 1 2 } ] , d n = d ( 1 υ n / c ) ( 1 υ n 2 / c 2 ) 1 2 [ ( 1 + Y 0 2 / c 2 ) 1 2 + ( Y 0 / W 0 ) { 1 ( 1 + W 0 2 / c 2 ) 1 2 } ] .
t 1 = d c ( 1 υ 1 / c ) ( 1 υ 1 2 / c 2 ) 1 2 [ 1 + W 0 / c ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ] = t ( 1 υ 1 / c ) ( 1 υ 1 2 / c 2 ) 1 2 [ 1 + W 0 / c ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ] , t 2 = t ( 1 υ 2 / c ) ( 1 υ 2 2 / c 2 ) 1 2 [ 1 + W 0 / c ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ] , t n = t ( 1 υ n / c ) ( 1 υ n 2 / c 2 ) 1 2 [ 1 + W 0 / c ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ] .
d 1 t 1 = d 2 t 2 = d n t n = c [ ( 1 + Y 0 2 / c 2 ) 1 2 + ( Y 0 / W 0 ) { 1 ( 1 + W 0 2 / c 2 ) 1 2 } ] [ 1 + W 0 / c ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ] ,
d 1 = d ( 1 υ 1 / c ) ( 1 υ 1 2 / c 2 ) 1 2 = d υ 1 ( d / c ) ( 1 υ 1 2 / c 2 ) 1 2 = d υ 1 t ( 1 υ 1 2 / c 2 ) 1 2 ,
x = x υ t ( 1 υ 2 / c 2 ) 1 2
t 1 = t ( 1 υ 1 / c ) ( 1 υ 1 2 / c 2 ) 1 2 = t ( υ 1 / c ) t ( 1 υ 1 2 / c 2 ) 1 2 = t υ 1 d / c 2 ( 1 υ 1 2 / c 2 ) 1 2 ,
t = t υ x / c 2 ( 1 υ / c 2 ) 1 2 ,
d 2 d 1 = d ( 1 υ 1 / c ) d ( 1 υ 2 / c ) [ 1 υ 2 2 / c 2 1 υ 1 2 / c 2 ] 1 2 ,
d 2 = d 1 [ 1 ( υ 2 υ 1 ) c ( 1 υ 1 υ 2 / c 2 ) { 1 ( υ 2 υ 1 ) 2 c 2 ( 1 υ 1 υ 2 / c 2 ) } ] ,
L s ( 1 υ 2 / c 2 ) 1 2 / ( υ 2 υ 1 ) ,
L s ( 1 υ 2 / c 2 ) / ( υ 2 υ 1 ) .
T = L s ( 1 υ 2 / c 2 ) υ 2 υ 1 L s ( 1 υ 2 / c 2 ) 1 2 W [ ( 1 υ 2 c 2 ) 1 2 ( 1 ( υ + W ) 2 c 2 ) 1 2 ] .
T = L s ( 1 υ 1 υ 2 / c 2 ) υ 2 υ 1 + L s c W 0 [ 1 ( 1 + W 0 2 c 2 ) 1 2 ] ,
υ 2 1 = L s T = 1 1 υ 1 υ 2 / c 2 υ 2 υ 1 + 1 c W 0 [ 1 ( 1 + W 0 2 c 2 ) 1 2 ] ,
υ 2 υ 1 1 υ 1 υ 2 / c 2 = υ 2 1 1 υ 2 1 [ 1 ( 1 + W 0 2 / c 2 ) 1 2 ] .
( υ 2 υ 1 ) / ( 1 υ 1 υ 2 / c 2 ) ,
υ 2 υ 1 1 υ 1 υ 2 / c 2 = υ 2 1 .
d 2 = d 1 ( 1 υ 2 1 / c ) ( 1 ( υ 2 1 ) 2 / c 2 ) 1 2 .
d 1 = d ( 1 υ / c ) ( 1 υ 2 / c 2 ) 1 2
α = z ( 1 x 2 ) 1 2 ( 1 ( x + z ) 2 ) 1 2 , z 2 / α 2 = ( 1 x 2 ) ( 1 [ x + z ] 2 ) = ( 1 x 2 ) ( 1 x 2 2 x z z 2 ) , z 2 [ 1 + α 2 ( 1 x 2 ) ] + 2 z x α 2 ( 1 x 2 ) α 2 ( 1 x 2 ) 2 = 0 , z = x α 2 ( 1 x 2 ) ± { x 2 α 4 ( 1 x 2 ) 2 + α 2 ( 1 x 2 ) 2 [ 1 + α 2 ( 1 x 2 ) ] } 1 2 1 + α 2 ( 1 x 2 ) = α ( 1 x 2 ) [ x α ± ( 1 + α 2 ) 1 2 ] ( ( 1 + α ) 1 2 x α ) ( ( 1 + α ) 1 2 + x α ) ,
= α ( 1 x 2 ) x α + ( 1 + α ) 1 2
W c = W 0 / c ( 1 υ 2 / c 2 ) W 0 υ / c 2 + ( 1 + W 0 2 / c 2 ) 1 2 .
L i = L s { ( 1 υ 2 / c 2 ) 1 2 ( Y / W ) [ ( 1 υ 2 / c 2 ) 1 2 ( 1 ( υ + W ) 2 / c 2 ) 1 2 ] } ( 1 ( υ + Y ) 2 / c 2 ) 1 2 ,
L i = L s [ ( 1 x 2 ) 1 2 ( y / z ) { ( 1 x 2 ) 1 2 ( 1 ( x + z ) 2 ) 1 2 } ) ] ( 1 ( x + y ) 2 ) 1 2 .
x + z = x + α ( 1 x 2 ) x α + ( 1 + α 2 ) 1 2 = α + x ( 1 + α 2 ) 1 2 x α + ( 1 + α 2 ) 1 2 ,
( 1 ( x + z ) 2 ) 1 2 = ( 1 x 2 ) 1 2 x α + ( 1 + α 2 ) 1 2
( 1 ( x + y ) 2 ) 1 2 = ( 1 x 2 ) 1 2 x β + ( 1 + β 2 ) 1 2
L i = L s [ ( 1 x 2 ) 1 2 y / z { ( 1 x 2 ) 1 2 ( 1 x 2 ) 1 2 / ( x α + ( 1 + α 2 ) 1 2 ) } ] ( 1 x 2 ) 1 2 / ( x β + ) 1 + β 2 ) 1 2 ) = L s { 1 β ( 1 x 2 ) x β + ( 1 + β 2 ) 1 2 α ( 1 x 2 ) x α + ( 1 + α 2 ) 1 2 [ 1 1 x α + ( 1 + α 2 ) 1 2 ] } ( x β + ( 1 + β 2 ) 1 2 ) = L s [ ( 1 + β 2 ) 1 2 + ( β / α ) ( 1 ( 1 + α 2 ) 1 2 ) ]
L i = L s [ ( 1 + Y c 2 / c 2 ) 1 2 + ( Y 0 / W 0 ) { 1 ( 1 + ( 1 + W 0 2 / c 2 ) 1 2 } ] .
T i = L s / c ( 1 + υ / c ) L / W ( 1 υ 2 / c 2 ) 1 2 [ ( 1 υ 2 / c 2 ) 1 2 ( 1 ( x + W ) 2 / c 2 ) 1 2 ] .
T i = L s c ( 1 + x ) L s / c z ( 1 x 2 ) 1 2 [ ( 1 x 2 ) 1 2 ( 1 ( x + z ) 2 ) 1 2 ] = L s c ( 1 + x ) L s c ( 1 x 2 ) ( 1 x 2 ) 1 2 ( 1 ( x + z ) 2 ) 1 2 z ,
T i = L s c ( 1 + x ) L s ( 1 x 2 ) [ 1 1 x α + ( 1 + α 2 ) 1 2 α ( 1 x 2 ) x α + ( 1 + α 2 ) 1 2 ] = L s c [ α + 1 ( 1 + α 2 ) 1 2 α ] = L s c [ W 0 / c + 1 ( 1 + W 0 2 / c 2 ) 1 2 W 0 / c ] .