8.2
Whence,
8.3
which is equivalent to 7.1.
It follows that if mₒ is the measured mass of Sₙ at rest, and a measured amount of work is done on Sₙ, producing a measured velocity of Sₙ, then the value of m in
8.4
will not be a constant necessarily, but will be
8.5
9. There has been nothing in this analysis that could fix the values of r and r′. Hence they must be classed as unknowns whose values might be determined for any specific object, Sₙ, by either an experimental determination of m for various conditions, or perhaps by a detailed analysis of the mathematical expressions for the work done on Sₙ under expressed conditions.
In actual experiment under many and varied conditions the value of m, (8.4), has been found to vary according to the equation
9.1
where V is the velocity of the electron, C is the velocity of light, and mₒ is the measured mass of the electron at rest or at negligible speeds.
Since this is equivalent to 8.S we have
9.2
and
9.3
as the experimentally determined values for the velocity-fraction and the internal-energy-fraction of acquired energy.
Then, if E is the work done on an electron at rest in a system, and mₒ is the measured mass of the electron at rest, the velocity-producing work done on the electron is r′E, and the velocity is given in 8.3.
10. One very important conclusion can be drawn from this analysis at once. It is
10.1 If the internal energy of any material particle remains a constant under all circumstances, its measured mass at all velocities will be constant.
And, since Sₙ may itself be considered as a material particle, (2.1), it follows that
10.2 If the measured mass of any material particle varies under certain variable conditions IT MUST BE COMPOSED OF SMALLER PARTICLES.
Naturally it must be definitely determined that the apparent variation of measured mass is not due to other factors such as hydrodynamic elements of a rigid particle moving at different speeds through a fluid. If these factors are completely accounted for and the measured mass still varies, it must be concluded that the particle is itself a system, Sₙ, made of still smaller particles.
If hydrodynamic factors are thoroughly accounted for in the measurement of the mass of an electron (and it is by no means certain they are) then we must conclude that THE ELECTRON IS NOT A SIMPLE PARTICLE BUT IS A SYSTEM OF STILL SMALLER PARTICLES.
A conclusion may be drawn concerning infinite mass. It has long been an objection to the experimental determination of the measured mass of the electron that if it reached the velocity of light it would have infinite mass. In everyday language, an electron with the speed of light would have more measured mass than the entire solar system plus all the other stars in the Milky Way! But now we may interpret infinite mass as follows:
10.3 A measured mass is infinite when r′, (8.1), is zero; whence, any object may be said to have infinite mass if and only if all work done on it is diverted to internal energy.
Conversely,
10.4 A measured mass is constant if and only if all work done on it produces velocity.
CONCLUSION
We have brought the Einsteinian and the Newtonian concepts of mass into one theoretical framework. There still remains one big question to be answered; Is all mass measured, or Einsteinian, energy-mass? Or is there some primal substance which may be called primal mass, which is present and remains constant throughout all changes of measured mass?
In other words, is mₒ, (4.3), a measured mass or a primal mass? There seems no way of answering that question at present. Since the measure of mass is purely relative and two masses may be equal, or measured equal, without examining the energy content of either, we have no way of finding an object which we can say has zero internal energy and comparing it with conventional objects.
In experiment, if we could find a particle whose mass is constant at all velocities we would still not know if it were a primal substance. We would only know that its energy content is constant—not whether it is zero or not.
Therefore, in this article the question of the basic nature of ultimate mass still remains unanswered. We are, however, in a position to define Newtonian mass and Einsteinian mass (or energy-mass) in common terms.
11.1 Newtonian mass:—Any material particle whose internal energy remains a constant under all conditions available.
11.2 Einsteinian mass:—Any material particle whose internal energy content varies under some conditions.
