Federalist 10

In the extent and proper structure of the Union, therefore, we behold a republican remedy for the diseases most incident to republican government. And according to the degree of pleasure and pride we feel in being republicans, ought to be our zeal in cherishing the spirit and supporting the character of Federalists. (10.23)

The last paragraph is a celebration of how the United States plans to address problems that republics face with solutions that aren't tyrannical. Because tyranny is what the British excel at—oooh, burn.

Instead of being plagued by the same problems that traditional republics have faced, as outlined by Hamilton in Federalist 9, the United States will be able to break with that history precisely because of its strong federal government. The very last sentence is a call-to-arms, urging state legislatures to support the Constitution as the best of solutions to America's ills.

Federalist 51

And happily for the REPUBLICAN CAUSE, the practicable sphere may be carried to a very great extent, by a judicious modification and mixture of the FEDERAL PRINCIPLE. (51.9)

There he goes with the all-caps again. Madison gets zero subtlety points for screaming FEDERAL PRINCIPLE.

But shouting aside, Federalist 51 actually ends in the same way that Federalist 10 does: by rejoicing in the fact that America is able to tweak a traditional republic for stability without losing any of its freedoms.

The repeat of this same sentence underlines the appeal they really needed to get: they needed to assure readers that they weren't trying to undermine US freedoms. Like, really really convince them.

The point was underlined, capitalized, then thrown into sky-high neon letters. After all, that was exactly what their opposition was accusing them of doing. By ending with the same refrain, they put extra stress on their dedication to preserving civil liberties, because they were definitely feeling the heat.