Synthesized answer
Elective kings must provide for succession to prevent the Commonwealth from dissolving upon their death [3]. If an elective king has no power to appoint his successor, and there is no other known person or assembly to elect a new one, then the Commonwealth dies with him and returns to a state of war [3]. This condition is contrary to the purpose for which monarchy was instituted, which was to avoid the "condition of a War of every man against every man" [1].
Furthermore, if there is no one who can give the sovereignty after the death of the first elected king, he is obligated by the law of nature to establish his successor [5]. This is to prevent those who entrusted him with the government from relapsing into the "miserable condition of Civill warre" [5]. The disposing of the successor is always left to the judgment and will of the present possessor [1].
Synthesized from the book passages below. Chat with the book on Feynman for follow-up.
From the book
that is in possession, has right to dispose of the Succession, or else that right is again in the dissolved Multitude. For the death of him that hath the Soveraign power in propriety, leaves the Multitude without any Soveraign at all; that is, without any Representative in whom they should be united, and be capable of doing any one action at all: And therefore they are incapable of Election of any new Monarch; every man having equall right to submit himselfe to such as he thinks best able to protect him, or if he can, protect himselfe by his owne sword; which is a returne to…
y their Authority, that the Election is made; and by the same it may (when the publique shall require it) be recalled. The Present Monarch Hath Right To Dispose Of The Succession The greatest difficultie about the right of Succession, is in Monarchy: And the difficulty ariseth from this, that at first sight, it is not manifest who is to appoint the Successor; nor many times, who it is whom he hath appointed. For in both these cases, there is required a more exact ratiocination, than every man is accustomed to use. As to the question, who shall appoint the Successor, of a Monarch that…
Democracie, or Aristocracie of another Common-wealth, Democratically, or Aristocratically governed, but Monarchically. And first, concerning an Elective King, whose power is limited to his life, as it is in many places of Christendome at this day; or to certaine Yeares or Moneths, as the Dictators power amongst the Romans; If he have Right to appoint his Successor, he is no more Elective but Hereditary. But if he have no Power to elect his Successor, then there is some other Man, or Assembly known, which after his decease may elect a new, or else the Common-wealth dieth, and dissolveth…
inheritance of his power; it is determined by his expresse Words, and Testament; or by other tacite signes sufficient. Succession Passeth By Expresse Words; By expresse Words, or Testament, when it is declared by him in his life time, viva voce, or by Writing; as the first Emperours of Rome declared who should be their Heires. For the word Heire does not of it selfe imply the Children, or nearest Kindred of a man; but whomsoever a man shall any way declare, he would have to succeed him in his Estate. If therefore a Monarch declare expresly, that such a man shall be his Heire, either…
e: For none have right to give that which they have not right to possesse, and keep to themselves, if they think good. But if there be none that can give the Soveraigntie, after the decease of him that was first elected; then has he power, nay he is obliged by the Law of Nature, to provide, by establishing his Successor, to keep those that had trusted him with the Government, from relapsing into the miserable condition of Civill warre. And consequently he was, when elected, a Soveraign absolute. Secondly, that King whose power is limited, is not superiour to him, or them that have the…