Wikipedia says that Hamilton confessed to adultery.
However, adultery is a crime, and a felony.
Why isn't he charged?
Statue of limitations?
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To address the frame challenge laid out in the comments, adultery does seem to have been illegal in some form or shape in Pennsylvania until 1973, though JoAnne Sweeny notes in "Undead Statutes: The Rise, Fall, and Continuing Uses of Adultery and Fornication Criminal Laws" that "the colonies of Pennsylvania and West Jersey did not appear to focus on crimes of sexual misconduct". Revolutionary Conceptions: Women, Fertility, and Family Limitation in America, by Susan E. Klepp, kind of echoes the sentiment though diverges on how it was enforced (around pp 237-238). In Pennsylvania specifically, she notes that the adultery laws were theoretically harsh on the one side, but that only 188 people (mostly women) got hauled in court for it from the 1690s to the 1790s on the other. (Men got the brunt of fornication charges.)
With this out of the way, in this specific case there doesn't seem to have been a plaintiff if the article on the affair that you've linked to is anything to go by.
Maria's husband, James Reynolds, basically seems to have abused the situation by extorting money from Hamilton. Not that two wrongs make a right or anything, but it doesn't look like the type of case that could end up in court. (If someone steals your stash of illicit substances, and you know the person who stole it, you're probably not going to seek remedy in court.) And even if it had gone to court, the complaint would have been against Maria more than Hamilton.
The article on Hamilton's wife, Eliza Schuyler, sketches out how she took it. Initially, she didn't believe the charges. When Hamilton confessed, she moved back with her parents for a while. As already noted, it was uncommon for wives to be the plaintiffs in adultery cases at the time. She eventually forgave him. The two reunited and went on to have another two children.