The First Prince was responsible for everything that happened in the Bridge World and the spies. It was a huge burden for him because Bridge World was an important place.
Meanwhile, the Second Prince was responsible for everything that happened to the internal military matter. The Second Prince was the one who managed the military base, from the soldiers' training until the soldier's allocation throughout the kingdom.
As for the Third Prince's responsibility was to help the Second Prince and the First Prince wherever he needed. He filled the empty spot.
Actually, the task to explore the ancient ruin was supposed to be given to the Third Prince. However, the Third Prince had currently had another important task at hand. He was assigned to exterminate and fend off the demonic monster. That was why the task for exploring the ancient ruin was given to the Second Prince.