Gittin, Chapter 8, Mishnah 9. Our Mishnah deals with two separate matters: the law of a man who divorces his wife and then spends the night with her after the divorce, and the law of one who marries a woman on the basis of a bald get.
"HaMegaresh es ishto v'lanah imo b'fundaki":
A man divorced his wife, and after the divorce he spent the night with her at an inn. This does not mean that they actually had marital relations, but that he secluded himself with her. Beis Shammai and Beis Hillel disagree about this law:
"Beis Shammai omrim: einah tzerichah heimenu get sheni" - no second get is required as a result of the seclusion at the inn, and we need not suspect that private seclusion constitutes an additional act of betrothal.
"U'Beis Hillel omrim: tzerichah heimenu get sheni" - she does require a second get, since in their view "hen hen eidei yichud hen hen eidei bi'ah": witnesses who see the two of them in a state of seclusion behind closed doors are considered witnesses to actual relations between them.
The further assumption is that a person does not intend his relations to be an act of promiscuity - he does not intend relations without the intent of marriage, and if it is within his power to do so, he will intend them for the sake of betrothal. It emerges, then, that we have witnesses to an act of betrothal, since relations are one of the ways of effecting betrothal. Once he closed himself in with his former wife, we require a second get from him, out of concern that they be regarded as having married a second time.
When does this apply?
The Mishnah qualifies this: "Eimasai - bizman shenisgarshah min hanisu'in" - that is, specifically where the first divorce came after they had lived together as husband and wife. However, "modim b'nisgarshah min ha'eirusin she'einah tzerichah heimenu get sheni, mipnei she'ein libo gas bah" - Beis Hillel agree with Beis Shammai that no second get is needed when the divorce was only from betrothal, because he is not close with her, and there is no reason to assume that relations took place between them. It is not self-evident here as it is in the case of a couple who already lived together as husband and wife.
A tied get and a bald get:
From here the Mishnah moves to an entirely different subject - the bald get. This matter relates to the tied get, a form of get that kohanim commonly used when divorcing. This get was especially difficult to write, so much so that Tosafos in Tractate Bava Basra note that the reason we do not make one nowadays is that we do not know exactly how to produce it. Making it involved writing a line or two of the get, folding them over, having a witness sign on the back of that fold, and then writing another line or two and having an additional witness sign, with at least three witnesses and perhaps even more. This get is called a tied get.
A bald get is a type of tied get, and it is "bald" in the sense that one of the folds has no witness on it. Every fold is supposed to bear a witness's signature on its back, and if one of the folds is missing a signature, this is a bald get. The concern with a bald get is that perhaps the husband appointed an additional person to testify, and that person did not sign, so the get is invalid. It is possible that in fact he appointed no one, but since such a concern exists, we invalidate a bald get.
The law of one who marries on the basis of a bald get:
"Kenasah b'get kere'ach - teitzei mizeh u'mizeh, v'chol hadrachim ha'eilu bah" - one who marries a woman on the basis of a bald get, this is an invalid get, and therefore she must leave both of them: she will need a divorce both from the second husband and from the first husband, and she may not live with either of them. And all the penalties listed in the previous Mishnayos apply to a woman who married on the basis of a bald get as well.
In summary: In this Mishnah we learned the dispute between Beis Shammai and Beis Hillel concerning one who divorces his wife and then spends the night with her at an inn - whether the seclusion requires a second get, and the qualification that this applies specifically where she was divorced from marriage and not from betrothal. We also clarified the nature of the tied get, the invalidity of the bald get that is missing a witness's signature on one of the folds, and the law of a woman who married on the basis of one - that she leaves both of them and all these penalties apply to her.