Please note that Mommsen uses the AUC chronology (Ab Urbe Condita), i.e. from the founding of the City of Rome. You can use this reference table to have the B.C. dates
From: The History of Rome, by Theodor Mommsen
Translated with the sanction of the author by William Purdie Dickson
Prerogatives of the Senate. The "Interregnum"
The prerogatives of this council of elders were based on the view that the rule over the community composed of clans rightfully belonged to the collective clan-elders, although in accordance with the monarchical principle of the Romans, which already found so stern an expression in the household, that rule could only be exercised for the time being by one of these elders, namely the king. Every member of the senate accordingly was as such, not in practice but in prerogative, likewise king of the community; and therefore his insignia, though inferior to those of the king, were of a similar character: he wore the red shoe like the king; only that of the king was higher and more handsome than that of the senator.
On this ground, moreover, as was already mentioned, the royal power in the Roman community could never be left vacant When the king died, the elders at once took his place and exercised the prerogatives of regal power. According to the immutable principle however that only one can be master at a time, even now it was only one of them that ruled, and such an "interim king" (-interrex-) was distinguished from the king nominated for life simply in respect to the duration, not in respect to the plenitude, of his authority. The duration of the office of -interrex- was fixed for the individual holders at not more than five days; it circulated accordingly among the senators on the footing that, until the royal office was again permanently filled up, the temporary holder at the expiry of that term nominated a successor to himself, likewise for five days, agreeably to the order of succession fixed by lot.
There was not, as may readily be conceived, any declaration of allegiance to the -interrex- on the part of the community. Nevertheless the -interrex- was entitled and bound not merely to perform all the official acts otherwise pertaining to the king, but even to nominate a king for life-- with the single exception, that this latter right was not vested in the first who held the office, presumably because the first was regarded as defectively appointed inasmuch as he was not nominated by his predecessor. Thus this assembly of elders was the ultimate holder of the ruling power (-imperium-) and the divine protection (-auspicia-) of the Roman commonwealth, and furnished the guarantee for the uninterrupted continuance of that commonwealth and of its monarchical--though not hereditarily monarchical--organization. If therefore this senate subsequently seemed to the Greeks to be an assembly of kings, this was only what was to be expected; it had in fact been such originally.
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Reference address : https://www.ellopos.net/elpenor/rome/1-05-original-constitution-rome.asp?pg=29