Not always the eldest. Scipio Africanus was Princeps Senatus when he was still quite young. To put into modern terms, it was the speaker, or president of the Senate; the Princeps gets to speak first and to lord it during Senatorial debates, but he still only gets one vote when push comes to shove. How much power he wields depends solely upon himself and how influential he is; the office only gives him a slight bump in prestige.
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