In The Prince, what does Machiavelli say is the best way to maintain the hereditary, mixed, civic, and ecclesiastical principalities?
Macchiavelli sees hereditary rule as the easiest type to maintain since the people are used to the customs and rule of their families, probably for many generations. There are less problems because the family has ruled for so long that any discontent has long been forgotten, and the people usually love the prince.
In a mixed principality, holding on to power is more difficult, but if the customs and language are the same, a ruler must make sure that the family of the former ruler has been killed and that the laws and taxes of the acquired country arent's changed. If the customs and language are different, then the ruler should either go live in the newly-acquired states or set them up as colonies, where they will be ruled by men the new ruler can trust. The ruler must guard against any other men coming from bordering states who might try to start trouble.
In a civil principality, a ruler is given power either by the people or the nobles. Machiavelli says it's more difficult to maintain power if the nobles have brought you to power because each of them feels he's as good as the new ruler and could foment a rebellion at any point. If a new ruler has the support of the people, he will have an easier time keeping his power because almost all of the people around him will be loyal.
Ecclesiastical principalities are the only ones that are "secure and happy" because their powers are derived from God, and it is God who maintains them.