Standing Rules of the United States Senate, Rule I
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Absence of the Vice President 
Paragraph 1 addresses the absence of the Vice President of the United States, who is under the United States Constitution the President of the Senate. The paragraph provides that the Senate shall choose a President pro tempore who will hold the office of President of the Senate should the Vice President be absent (the Vice President usually only presides over the Senate on ceremonial occasions). The President pro tempore shall execute "the duties thereof during the pleasure of the Senate" until another is elected or his term of office as a Senator expires.
Acting President pro tempore 
Paragraph 2 addresses the absence of the Vice President pending election of a President pro tempore. It states that the Assistant Secretary of the Senate shall perform all duties of Acting President pro tempore or Secretary of the United States Senate.
Rights of President pro tempore 
Paragraph 3 addresses the rights of President pro tempore. It states that the President pro tempore has the right to name a Senator to perform the duties of the Chair if absent and in writing. This duty includes the signing of duly enrolled bills and joint resolutions. This duty, however, does not extend beyond an adjournment. The exception is by unanimous consent and the Senator named has the right to name "in open session, or, if absent, in writing, a Senator to perform the duties of the Chair, but not to extend beyond an adjournment, except by unanimous consent."