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  1. For the following energy, pressure, and mass relations see H. E. Ives, J. Opt. Soc. Am. 32, 32 (1942).
    [Crossref]
  2. The formula for any interval between two velocities v1 and v2, derived by the same reasoning as above isf(v1, v2)=[c2(g2-g1)]/(v2g2-v1g1).If we put f(v1v2)=v1+[(v2−v1)/2], and put v1=v2/2, we have for g1, in term of v2,g1=1/[1-(v22/8c2)].Putting this in the above expression and solving for g2, we getg2=1/[1-12(v22/c2)-332(v24/c4)].Repeating this process for successive equal increments we get values for the coefficient of v4/c4, of 19, 15/128, 3/25, etc., which asymptotically approach the value 18; and values for the coefficient of v6/c6 which similarly approach 116.
  3. See F. K. Richtmyer, Introduction to Modern Physics (McGraw-Hill, New York, 1934), second edition, p. 723.
  4. If we assume, contrary to our result, the M is invariant with velocity we would have for the conservation of momentumE(cΔt)c2c+E(cΔt)m′[c-f(v)c+f(v)]c=Mvwhere m′ is a new (variable) coefficient of mass equivalence of radiation. The conservation of mass would then be expressed byE(cΔt)c2+M=E(cΔt)m′[c-f(n)c+f(v)]+M,givingm′=1c2[c-f(v)c+f(v)]so thatMv=2E(cΔt)c2c.The kinetic energy of the particle corresponding to momentum Mv is 12Mv2, giving12Mv2=E(cΔt)vc,which obviously is not equal to E(cΔt) [1-c-f(v)c+f(v)] ; so that conservation of energy and momentum would not hold, in the impact process here studied, for an invariant M.
  5. A. S. Eddington, Mathematical Theory of Relativity (Cambridge University Press, London, 1937), second edition, p. 31.
  6. F. K. Richtmyer, reference 3, p. 719.
  7. Of interest in this general connection is an article by P. S. Epstein [Am. J. Phys. 10, 1 (1942)] in which it is pointed out that, given the variation of mass with velocity here deduced, the variation of clock rate may be derived. It thus appears probable that all the behaviors characteristic of the restricted theory of relativity may be obtained from earlier principles.
    [Crossref]

1942 (2)

For the following energy, pressure, and mass relations see H. E. Ives, J. Opt. Soc. Am. 32, 32 (1942).
[Crossref]

Of interest in this general connection is an article by P. S. Epstein [Am. J. Phys. 10, 1 (1942)] in which it is pointed out that, given the variation of mass with velocity here deduced, the variation of clock rate may be derived. It thus appears probable that all the behaviors characteristic of the restricted theory of relativity may be obtained from earlier principles.
[Crossref]

Eddington, A. S.

A. S. Eddington, Mathematical Theory of Relativity (Cambridge University Press, London, 1937), second edition, p. 31.

Epstein, P. S.

Of interest in this general connection is an article by P. S. Epstein [Am. J. Phys. 10, 1 (1942)] in which it is pointed out that, given the variation of mass with velocity here deduced, the variation of clock rate may be derived. It thus appears probable that all the behaviors characteristic of the restricted theory of relativity may be obtained from earlier principles.
[Crossref]

Ives, H. E.

Richtmyer, F. K.

See F. K. Richtmyer, Introduction to Modern Physics (McGraw-Hill, New York, 1934), second edition, p. 723.

F. K. Richtmyer, reference 3, p. 719.

Am. J. Phys. (1)

Of interest in this general connection is an article by P. S. Epstein [Am. J. Phys. 10, 1 (1942)] in which it is pointed out that, given the variation of mass with velocity here deduced, the variation of clock rate may be derived. It thus appears probable that all the behaviors characteristic of the restricted theory of relativity may be obtained from earlier principles.
[Crossref]

J. Opt. Soc. Am. (1)

Other (5)

The formula for any interval between two velocities v1 and v2, derived by the same reasoning as above isf(v1, v2)=[c2(g2-g1)]/(v2g2-v1g1).If we put f(v1v2)=v1+[(v2−v1)/2], and put v1=v2/2, we have for g1, in term of v2,g1=1/[1-(v22/8c2)].Putting this in the above expression and solving for g2, we getg2=1/[1-12(v22/c2)-332(v24/c4)].Repeating this process for successive equal increments we get values for the coefficient of v4/c4, of 19, 15/128, 3/25, etc., which asymptotically approach the value 18; and values for the coefficient of v6/c6 which similarly approach 116.

See F. K. Richtmyer, Introduction to Modern Physics (McGraw-Hill, New York, 1934), second edition, p. 723.

If we assume, contrary to our result, the M is invariant with velocity we would have for the conservation of momentumE(cΔt)c2c+E(cΔt)m′[c-f(v)c+f(v)]c=Mvwhere m′ is a new (variable) coefficient of mass equivalence of radiation. The conservation of mass would then be expressed byE(cΔt)c2+M=E(cΔt)m′[c-f(n)c+f(v)]+M,givingm′=1c2[c-f(v)c+f(v)]so thatMv=2E(cΔt)c2c.The kinetic energy of the particle corresponding to momentum Mv is 12Mv2, giving12Mv2=E(cΔt)vc,which obviously is not equal to E(cΔt) [1-c-f(v)c+f(v)] ; so that conservation of energy and momentum would not hold, in the impact process here studied, for an invariant M.

A. S. Eddington, Mathematical Theory of Relativity (Cambridge University Press, London, 1937), second edition, p. 31.

F. K. Richtmyer, reference 3, p. 719.

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Equations (35)

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ν = ν 0 ( c - v c + v ) ,
E = E 0 ( c - v c + v ) 2 ,
l = l 0 ( c + v c - v ) = c Δ t ( c + v c - v ) ,
E l = E 0 ( c Δ t ) ( c - v c + v ) .
E ( c Δ t ) [ c - f ( v ) c + f ( v ) ] .
E [ c - f ( v ) ] - E [ c - f ( v ) c + f ( v ) ] 2 [ c + f ( v ) ] = P f ( v )
P = 2 E [ c - f ( v ) c + f ( v ) ] .
Pressure = change of momentum time = change of ( mass × velocity ) time .
2 E [ c - f ( v ) c + f ( v ) ] c Δ t c - f ( v ) = m E ( c Δ t ) c + m E ( c Δ t ) [ c - f ( v ) c + f ( v ) ] c
m = 1 c 2 .
E ( c Δ t ) c 2
E ( c Δ t ) c 2 [ c - f ( v ) c + f ( v ) ] .
E ( c Δ t ) c 2 c = E ( c Δ t ) c
E ( c Δ t ) c 2 [ c - f ( v ) c + f ( v ) ] c = E ( c Δ t ) c [ c - f ( v ) c + f ( v ) ] .
E ( c Δ t ) c 2 + M ,
E ( c Δ t ) c 2 [ c - f ( v ) c + f ( v ) ] + M .
E ( c Δ t ) c 2 + M = E ( c Δ t ) c 2 [ c - f ( v ) c + f ( v ) ] + M g ,
E ( c Δ t ) c = - E ( c Δ t ) c [ c - f ( v ) c + f ( v ) ] + M g v .
f ( v ) = c 2 ( g - 1 ) g v .
g = 1 1 - ( v 2 / 2 c 2 ) .
g = 1 1 - ( v 2 / 2 c 2 ) - 1 8 ( v 4 / c 4 - )
g = 1 [ 1 - ( v 2 / c 2 ) ] 1 2 ,
M 0 v [ 1 - ( v 2 / c 2 ) ] 1 2
M c 2 [ 1 ( 1 - v 2 c 2 ) 1 2 - 1 ] .
E ( c Δ t ) [ 1 - ( c - f ( v ) ) ( c + f ( v ) ) ]
E ( c Δ t ) [ 1 - c - f ( v ) c + f ( v ) ] = M c 2 [ 1 ( 1 - v 2 c 2 ) 1 2 - 1 ] = M c 2 [ g - 1 ] .
E ( c Δ t ) c 2 + M 0 = E ( c Δ t ) c 2 [ c - f ( v ) c + f ( v ) ] + M 0 g
f(v1,v2)=[c2(g2-g1)]/(v2g2-v1g1).
g1=1/[1-(v22/8c2)].
g2=1/[1-12(v22/c2)-332(v24/c4)].
E(cΔt)c2c+E(cΔt)m[c-f(v)c+f(v)]c=Mv
E(cΔt)c2+M=E(cΔt)m[c-f(n)c+f(v)]+M,
m=1c2[c-f(v)c+f(v)]
Mv=2E(cΔt)c2c.
12Mv2=E(cΔt)vc,