Synthesized answer
If others elect a successor, the right to do so can either be held by the present monarch if they have the right to dispose of the succession, or it returns to the "dissolved Multitude" [1]. Elective Kings and Princes do not hold sovereign power in proprietorship, but only in use [1].
In the case of an elective king whose power is limited to his life and who has no power to elect his successor, if there is another person or assembly known to elect a new ruler, sovereignty was in them before [4]. If there is no one known to have the power to give sovereignty after the decease of the elected ruler, the commonwealth dissolves, and sovereignty returns to the condition of war [4].
Synthesized from the book passages below. Chat with the book on Feynman for follow-up.
From the book
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…
s in a person subject, and may be assumed by the Soveraign at his pleasure; and consequently the Right is in himselfe. And if it be in no particular man, but left to a new choyce; then is the Common-wealth dissolved; and the Right is in him that can get it; contrary to the intention of them that did institute the Common-wealth, for their perpetuall, and not temporary security. In a Democracy, the whole Assembly cannot faile, unlesse the Multitude that are to be governed faile. And therefore questions of the right of Succession, have in that forme of Government no place at all. In an…
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…
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…
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…