Why do the Elucidated Brethren want to summon a dragon in the novel, Guards! Guards!?

The Elucidated Brethern long for the days when Ankh-Morpork was ruled by a king rather than by the Patrician and the guilds. They believe that the royal line has not died out and believe the legends and prophesies that say that the king will return. The rightful heir can prove his claim to the throne by killing a dragon, so they summon the dragon so the king can be revealed. However, many of them have selfish, ulterior motives for wanting to be in control of a large, fire-breathing monster.