The first, in 1893, was his "displacement law." He saw that the frequency (or wavelength) with the maximum intensity in the light spectrum shifts (is displaced) toward higher frequencies as the temperature is increased.

The displacement law is most often written in terms of wavelength λ as

But it is perhaps more intuitive to write it in terms of the maximum frequency ν_max

or even write it with the Planck constant and the Boltzmann constant as

This shows clearly that in thermal equilibrium the most common photon energy hν is simply related to kT, the average energy. As the temperature goes up, the frequency and energy of the radiation goes up.

Wien could see that this greatly resembles the Maxwell-Boltzmann distribution of gas particles. The velocity v with the most particles, the maximum in the Maxwell-Boltzmann distribution, is when the energy of the particles is near to the average energy per particle kT.

½kT is the energy in each of the three degrees of freedom, the three components of the velocity in three dimensions.

So Wien knew that when the temperature increases, the velocities of gas particle increase. That may have led Wien to see that the radiation frequencies increase with temperature, and that there might be a fundamental relation between the energy in the matter and energies in radiation. The most important relation may be that in equilibrium energy is delivered equally to matter and radiation, the idea of equipartition of the energy.

Note the resemblance between the distribution of blackbody radiation as a function of temperature and the Maxwell-Boltzmann distribution of velocities.

The similarity between the two is the rise to a maximum with a power law on one side and an exponential decline on the other. The difference is 1) because the radiant energy (the number of photons) increases greatly as temperature goes up, but the number of molecules is held constant; 2) because molecules are plotted against velocity, not energy (½mv²), and 3), the velocity increases to the right.

Wien knew of this resemblance and suggested that the radiation frequencies are distributed in somewhat the same fashion as the molecular velocities,

He proposed this formula for the distribution of radiation, now known as Wien's law.

where a and b are arbitrary constants. The exponential decline of radiation as frequency increases clearly follows the exponential decline in gas particle energies and more importantly is the "Boltzmann factor" e^-E/kT that reduces the "statistical weight" of high energy states in Boltzmann's statistical mechanics.

Wien was carefully following the experimental evidence for the distribution of energy as a function of frequency or wavelength and knew that his shape was a very good fit to the experimental data. Many others proposed various formulas, but Wien's was the only one with a theory behind it. The others were just ad hoc fits to the data.