We are up to Mishnah 4 in the first chapter of Maseches Sotah. The Mishnah describes what takes place when the husband and the suspected wife arrive in Jerusalem, and the procedure designed to bring the woman to admit her sin before she drinks the sotah waters.
The words of the Mishnah:
"Vaya'alu osah l'Beis Din HaGadol sheb'Yerushalayim" - they bring her up to the Great Court, that is, the Sanhedrin that sits in Jerusalem, in the Beis HaMikdash.
"U'me'ayemin aleha" - they frighten her and impress upon her the gravity of the situation.
"K'derech she'me'ayemin al edei nefashos" - in the same manner that the court warns witnesses who come to testify in capital cases.
Just as the court tells witnesses in capital cases: if you wish to testify against a man in a way that will bring about his death, you must understand how serious this is - as the Mishnah in Maseches Sanhedrin describes the methods of warning them, so that they will not say anything untrue - so too they say to the woman: if you really did commit adultery, it is better for you to admit it than to place yourself in a situation of mortal danger, for she is liable to die from drinking the sotah waters.
"V'omrim lah: Biti":
They open with warm, endearing language - "my daughter" - and explain to her that many things could have brought her to sin. By pointing to external factors, there is hope that she will be willing to admit the truth:
"Harbeh yayin oseh" - drinking wine, and under its influence she may have done what she should not have done.
"Harbeh s'chok oseh" - excessive laughter and light-headedness can lead to this.
"Harbeh yaldus oseh" - immaturity may have been the cause.
"Harbeh sh'cheneha hara'im osim" - bad neighbors who influenced her for the worse, and she herself is not a bad person. Things happen.
"Aseh lishmo hagadol shenichtav bikdushah, shelo yimacheh al hamayim":
They continue and ask her to admit the truth for the sake of the great Name of Hashem, which is written in holiness and should not be erased into the water. For the passage of sotah is written on parchment, and that writing - including the Name of Hashem - is erased into the water that she drinks. True, it is permitted to do so, since the Torah itself commands it, but it is proper to avoid it if possible. Therefore they say to her: admit it, and do not force us to erase His Name.
"V'omrim lifaneha devarim she'einam k'dai l'shomah":
They say before her things that are not fitting or appropriate for her to hear - "hi v'chol mishpachas beis aviha" - neither for her nor for her family. The Gemara explains that they tell her about the deeds of the great men of Israel, who admitted their sins and were not ashamed, such as the incident of Yehudah and Tamar. Indeed it is not appropriate to share such matters with her, and nevertheless they do so in the hope that it will influence her to admit her sin.
In summary: In this Mishnah we learned that the suspected woman is brought up to the Great Court in Jerusalem and is warned in the same manner that witnesses in capital cases are warned, so that she will admit the truth and not place herself in mortal danger. We saw the methods of persuasion: pointing to the external factors that can lead to sin (wine, laughter, immaturity and bad neighbors), the appeal for the sake of His great Name that it not be erased into the water, and saying before her things that are not fitting for her to hear - the deeds of the great men of Israel who admitted their sins, such as the incident of Yehudah and Tamar.