Casimir heat transfer (Ballistic heat transfer) paper

[Sciencestd]

In general, in the paper Casimir assumes that between the boundaries of a crystal the heat transfer can be imagined as electromagnetic radiation between two hollow bodies connected by a tube (at least this what I understood from his approach),

I'm reading about Casimir limit (Ballistic heat transfer), so I downloaded his origin paper. I don't know where it comes from the first equation and how he goes from one to another, from equation 1 to 2 to 3 (shown in the pic above)...One can download the paper from: https://www.sciencedirect.com/science/article/pii/S0031891438801622

 

[Valerie Park]

In this paper, Casimir considers the problem of heat transfer between two bodies, separated by an extended gap. He assumes that the heat transfer can be represented as a radiation field propagating between the two bodies. He then derives an expression for the power flux through the gap, and shows that it is proportional to the fourth power of the temperature difference between the two bodies. This is known as the Casimir limit, and is valid when the gap is much larger than the wavelength of the radiation. In the equation shown in the picture, the left-hand side represents the power flux, and the right-hand side represents the temperature difference between the two bodies.