#### TL;DR
This paper introduces Einstein's now famous concept o...

This is the previous investigation Einstein is referring to: [On th...

The Maxwell-Hertz equations provide a mathematical description of e...

The term in the denominator: $$\sqrt{1 - \frac{v^2}{V^2}}$$ arises...

This difference has a simple physical interpretation: the body’s ki...

This tells us that the kinetic energy of the body changes when ligh...

Question here: Why is it that the difference in kinetic energy must...

> The mass of a body is a measure of its energy-content; if the ene...

Question here: Why is it that the difference in kinetic energy must be explained by a change in mass, rather than a change in velocity?
The term in the denominator: $$\sqrt{1 - \frac{v^2}{V^2}}$$ arises due to time dilation, a core concept of special relativity. It reflects how energy and time are altered by the motion of the observer relative to the speed of light.
The Maxwell-Hertz equations provide a mathematical description of electromagnetic waves, including light, which Einstein uses to explore how light interacts with a moving body.
> The mass of a body is a measure of its energy-content; if the energy changes by L, the mass changes in the same sense by $L/9 × 10^{20}$, the energy being measured in ergs, and the mass in grammes.
This difference has a simple physical interpretation: the body’s kinetic energy in one frame differs from its kinetic energy in the other frame by an amount related to the emitted light.
This tells us that the kinetic energy of the body changes when light is emitted, and the amount of this change depends on the speed of the moving frame and the energy of the emitted light.
#### TL;DR
This paper introduces Einstein's now famous concept of mass-energy equivalence $E=mc^2$. It's the first time that mass and energy are not separate entities but are instead deeply linked.
Einstein’s approach in the paper revolves around the concept that if an object emits energy in the form of radiation then its mass should decrease - and if it absorbs energy its mass should increase.
Einstein’s derivation shows that even a small emission of light from a body leads to a corresponding change in its mass. He shows that the change in mass (\( \Delta m \)) is related to the change in energy (\( \Delta E \)) through the speed of light squared: $$\Delta m = \frac{\Delta E}{c^2}$$
You can find an html version of this paper here: [Does the Inertia of a Body Depend upon its Energy-Content?](https://www.fourmilab.ch/etexts/einstein/E_mc2/www/)
This is the previous investigation Einstein is referring to: [On the Electrodynamics of Moving Bodies](https://www.fourmilab.ch/etexts/einstein/E_mc2/www/)