"Immediately after they shall be assembled in Consequence of the first Election, they shall be divided as equally as may be into three Classes. The Seats of the Senators of the first Class shall be vacated at the Expiration of the second Year, of the second Class at the Expiration of the fourth Year, and of the third Class at the Expiration of the sixth Year, so that one third may be chosen every second Year; and if Vacancies happen by Resignation, or otherwise, during the Recess of the Legislature of any State, the Executive thereof may make temporary Appointments until the next Meeting of the Legislature, which shall then fill such Vacancies."
Comment: On April 6, 1789, the Senate reached a quorum. On May 13, the Senate then divided themselves into three (3) classes. Senators were then separated into three geographically balanced classes. However, Senators from the same State were in different classes. To create the staggered terms of the Senate, three equal sized pieces of paper numbered 1, 2, and 3 rolled up and placed in a box. The Senator drawing "1" would end his term at the end of two years. The Senator drawing "2" would end his term at the end of the fourth year. And, the Senator drawing a "3," his term would end on the sixth year.
The last section of the clause grants the State's governor the authority to make Senate appointments in the event there was a vacancy and the legislature was not in session.
Primary Source. The Heritage Guide to the Constitution.
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