Introduction:
In this scenario, we have a square made of four thin rods of the same material and dimensions. Two diagonally opposite corners of the square are at temperatures of 40 degrees Celsius and 10 degrees Celsius. We need to determine the temperature difference between the other two corners of the square when only heat conduction takes place.

Heat Conduction:
Heat conduction is the transfer of heat energy within a material or between different materials in direct contact. It occurs due to the random movement of atoms and molecules, where higher energy particles transfer their energy to lower energy particles. In this case, heat conduction will occur between the rods in the square.

Heat Transfer:
Heat transfer occurs from the hotter end to the colder end until thermal equilibrium is reached. In this scenario, heat will flow from the 40 degrees Celsius corner to the 10 degrees Celsius corner through the rods.

Assumptions:
To simplify the calculation, we assume that the rods have uniform heat conductivity and that there is no heat loss to the surroundings.

Analysis:
We can analyze this scenario using the concept of thermal equilibrium. When the system reaches thermal equilibrium, the temperature will be the same throughout the square. Therefore, the temperature difference between the other two corners will be the same.

Calculating the temperature difference:
To calculate the temperature difference between the other two corners, we can consider the heat conduction through the rods as a series of small temperature differences. The heat flow will be proportional to the temperature difference across each rod.

Using Fourier's Law:
We can use Fourier's Law of heat conduction to calculate the heat flow through the rods. Fourier's Law states that the rate of heat transfer through a material is proportional to the temperature difference across it and inversely proportional to the material's thermal conductivity.

Conclusion:
In conclusion, the temperature difference between the other two corners of the square will depend on the heat conduction through the rods. By considering the principles of heat conduction and using Fourier's Law, we can calculate the temperature difference accurately.