Firstly, Madison notes that the federal government would have the power to regulate intellectual property; that is, copyright, trademarks, and patents. He thinks this is obviously a good idea, and doesn't dwell too long on it. He's right: having each state enforce its own patents and such would be a nightmare. Some system of intellectual property rights must exist to encourage development, so it should probably be a unified federal system.
Next, he examines the power to create a federal district and exercise jurisdiction over it. This power, in a few years' time, would result in the establishment of Washington, D.C. The Constitution doesn't specify where the federal district would go, and the debate over the location of the new capital would get quite heated. Madison and the other authors of the Constitution rightly recognized that having the capital in a state would give that state too much power over the federal government, which was in those days quite weak. What no one at the time considered was the possibility that the capital district would get so populous that it would agitate for its own representation as a state. Despite many proposals in the 200 years since, this problem remains unresolved.The third provisions Madison examines deals with treason, and how to punish it. If treason is against the United States, Madison reasons, the United States must be able to punish it. But limits are important. Under the British system, all sorts of anti-administration activity was classed as treason, and in the rest of Europe it was even more egregious. As he writes, "as new-fangled and artificial treasons have been the great engines by which violent factions ... by inserting a constitutional definition of the crime, fixing the proof necessary for conviction of it, and restraining the Congress, even in punishing it, from extending the consequences of guilt beyond the person of its author." This ensures that even if one administration or other wishes to punish its enemies, treason will be unavailable as an accusation except for serious crimes against the nation. This provision likely kept the United States from degenerating into a republic where changes in government are accompanied by bloodshed.
Next, Madison discusses the power to admit new states to the Union. This seems obvious now, but at the time it was unclear whether and how new states were to be formed from the western lands or elsewhere. The Confederation government had taken the lead by erecting new territories in the Old Northwest, but this provision made certain that any state formed there would be admitted and treated the same as the original thirteen. Along the same lines is a fifth power, the power to govern the federal territories that are not yet states. This was not terribly controversial at the time, but would become so in the days leading up to the Civil War. Since then, it has been uncontroversial once more.The sixth power Madison discusses is the so-called Republican Guaranty clause, or the power to ensure that every state has a republican form of government. This made anti-Federalists nervous: what if the federal government decided that their state was insufficiently republican? Would the feds be able to rewrite the state's constitution to its liking? This was troubling since then, as now, no one knew exactly what this clause meant. All the states had different voting rights, different labor rights (i.e., slavery) and so on. Were they all equally republican?
Madison seeks to set the people's minds at ease: the guaranty, he says, "supposes a pre-existing government of the form which is to be guaranteed." As long as the keep up the same degree of a republic as they had in 1788, there can be no complaint from the feds, right? As long as the states have some sort of a republic, as they currently understood the term, there was no problem. Only the introduction of "aristocratic or monarchial innovations" would violate the clause. That's not a terribly exact explanation, and while it came up from time to time, no one ever really figured out what it meant.
Seventh, he discusses the provision that says the debts the United States contracted before the Constitution will still be valid afterward. Basically: we'll still pay our bills if this thing passes. This should've kept bankers from being opposed to the Constitution, anyway.
Eighth, he briefly mentions the amendment process. As much as the Publius authorship group likes the Constitution, they know it's not perfect and that defects may emerge later, as they did under the Articles. If so, and if enough people think it's a problem, it can be fixed. We've done it twenty-eight times since then and, while it's not easy, it's a much better solution than having revolutions all the time like France.
Lastly, Madison talks about the ratification process. Under the Articles, all thirteen states had to approve a change. The Constitution, however, says it will become effective if nine states ratify it. What's up with that? Madison recognizes that these are delicate questions, but reminds his readership that in the history of the Articles of Confederation, never have all thirteen states agreed on anything. The question, he says "is answered at once by recurring to the absolute necessity of the case; to the great principle of self-preservation; to the transcendent law of nature and of nature's God, which declares that the safety and happiness of society are the objects at which all political institutions aim, and to which all such institutions must be sacrificed." Something had to give, and constitution-making is always a revolutionary act. Did the Framers cut a few corners? Sure, but desperate times call for desperate measures.

