Page:LewisRevision.djvu/11

testing this point would be very great, of course, but perhaps not insurmountable.

The plausibility of our fundamental assumption which led directly to equation (15) has been greatly increased by the agreement, between this equation and Kaufmann's results, and also perhaps by the similarity between this equation and those deduced from electromagnetic theory. But the simplest as well as the strongest evidence of the correctness of our point of view comes from a consideration of the non-Newtonian equation for kinetic energy.

The integration of equation (14) obviously does not yield the simple Newtonian equation,

$\mathrm {E} '=1/2mv^{2}$.

This equation must be replaced by one that is obtained as follows : —

Let a body, which at rest has the mass m₀, be given the velocity v. As its internal energy changes, its mass will change according to equation (7), and

$m-m_{0}={\frac {\mathrm {E} '}{\mathrm {V} ^{2}}}$

where E' is the acquired kinetic energy and m—m/m₀ is the increase in mass.

Eliminating m₀ between this equation and (15) gives

$\mathrm {E} '=m\mathrm {V} ^{2}\left[1-\left(1-\beta ^{2}\right)^{1/2}\right]$.

(16)

This is the general formula for the kinetic energy of a moving body. As usual β represents v/V, the ratio of this velocity to the velocity of light.

From equations (10), (15), and (16) may be constructed the whole science of non-Newtonian dynamics, of which Newtonian dynamics furnishes a limiting case, namely, for velocities which are negligible in comparison with the velocity of light.

For example, expanding (16) by the binomial theorem gives

$\mathrm {E} '=m\mathrm {V} ^{2}\left({\frac {1}{2}}\beta ^{2}+{\frac {1}{8}}\beta ^{4}\dots \right)$.

(17)

For low values of β the higher terms may be neglected and

$\mathrm {E} '={\frac {1}{2}}mv^{2}$

That is, the limit approached by the kinetic energy of a body as the velocity approaches zero is, as in ordinary mechanics, one half the product of the mass and the square of the velocity. At the other limit of velocity when β=1, it follows from (16) that

$\mathrm {E} '=m\mathrm {V} ^{2}$.

(18)