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

The formula for any interval between two velocities ν_{1} and ν_{2}, derived by the same reasoning as above is ƒ(ν_{1}, ν_{2}) = [c^{2}(g_{2}g_{1})]/(ν_{2}g_{2}ν_{1}g_{1}). If we put ƒ(ν_{1}ν_{2}) =ν_{1}+[(ν_{2}ν_{1})/2], and put ν_{1}=ν_{2}/2, we have for g_{1}, in term of ν_{2}, g_{1}=1/[1(ν_{2}^{2}/8C^{2})]. Putting this in the above expression and solving for g_{2}, we get g_{2}=1/[1½(ν_{2}^{2}/c^{2})3/32(ν_{2}^{4}/c^{4})]. Repeating this process for successive equal increments we get values for the coefficient of ν^{4}/c^{4}, of 1/9, 15/128, 3/25, etc., which asymptotically approach the value ⅛; and values for the coefficient of ν^{6}/c^{6} which similarly approach 1/16

See F. K. Richtmyer, Introduction to Modern Physics (McGrawHill, New York, 1934), second edition, D. 723.

If we assume, contrary to our result, the M is invariant with velocity we would have for the conservation of momentum [Equation] where m′ is a new (variable) coefficient of mass equivalence of radiation. The conservation of mass would then be expressed by [Equation] giving [Equation] so that [Equation] The kinetic energy of the particle corresponding to momentum Mν is ½Mν^{2}, giving [Equation] which obviously is not equal to [Equation]; 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.

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.
Eddington, A. S.
A. S. Eddington, Mathematical Theory of Relativity (Cambridge University Press, London, 1937), second edition, p. 31.
Ives, H. E.
For the following energy, pressure, and mass relations see H. E. Ives, J. Opt. Soc. Am. 32, 32 (1942).
Richtmyer, F . K.
F . K. Richtmyer, reference 3, p. 719.
Richtmyer, F. K.
See F. K. Richtmyer, Introduction to Modern Physics (McGrawHill, New York, 1934), second edition, D. 723.
Other
For the following energy, pressure, and mass relations see H. E. Ives, J. Opt. Soc. Am. 32, 32 (1942).
The formula for any interval between two velocities ν_{1} and ν_{2}, derived by the same reasoning as above is ƒ(ν_{1}, ν_{2}) = [c^{2}(g_{2}g_{1})]/(ν_{2}g_{2}ν_{1}g_{1}). If we put ƒ(ν_{1}ν_{2}) =ν_{1}+[(ν_{2}ν_{1})/2], and put ν_{1}=ν_{2}/2, we have for g_{1}, in term of ν_{2}, g_{1}=1/[1(ν_{2}^{2}/8C^{2})]. Putting this in the above expression and solving for g_{2}, we get g_{2}=1/[1½(ν_{2}^{2}/c^{2})3/32(ν_{2}^{4}/c^{4})]. Repeating this process for successive equal increments we get values for the coefficient of ν^{4}/c^{4}, of 1/9, 15/128, 3/25, etc., which asymptotically approach the value ⅛; and values for the coefficient of ν^{6}/c^{6} which similarly approach 1/16
See F. K. Richtmyer, Introduction to Modern Physics (McGrawHill, New York, 1934), second edition, D. 723.
If we assume, contrary to our result, the M is invariant with velocity we would have for the conservation of momentum [Equation] where m′ is a new (variable) coefficient of mass equivalence of radiation. The conservation of mass would then be expressed by [Equation] giving [Equation] so that [Equation] The kinetic energy of the particle corresponding to momentum Mν is ½Mν^{2}, giving [Equation] which obviously is not equal to [Equation]; 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.
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.
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