Maseches Yevamos, Chapter 1, Mishnah 4. This Mishnah presents an opinion that disputes the principle established in the opening Mishnah of the chapter, namely, that the rival wife of an ervah is exempt from yibum.
"Beis Shammai matirin ha'tzaros la'achin" - Beis Shammai permit the rival wives to the brothers:
A brother's wife is forbidden by Torah law, and it is the mitzvah of yibum that permits her. The ervah herself is exempt from yibum because the prohibition of ervah upon her is added to her status as a brother's wife, and therefore yibum does not permit her. Her rival wife, however, according to Beis Shammai, is obligated in the mitzvah of yibum and may marry the brother, since in their view there is no such concept as a rival wife of an ervah being exempt from yibum. Beis Hillel, as emerges from the opening Mishnah, forbid her to the brother and exempt her from yibum.
Two practical consequences of the dispute:
"Chaltzu" - a rival wife of an ervah who received chalitzah from her yavam: "Beis Shammai poslin min ha'kehunah, u'Beis Hillel machshirin" - Beis Shammai disqualify her from marrying into the kehunah, and Beis Hillel declare her fit. According to Beis Shammai this chalitzah is a valid chalitzah, since the woman is obligated either in yibum or in chalitzah, and consequently she is a full chalutzah - and a chalutzah is forbidden to a kohen. Beis Hillel hold that this chalitzah has no significance whatsoever, since she was obligated neither in yibum nor in chalitzah, and therefore she remains permitted to a kohen.
"Nisu" - a rival wife of an ervah who actually underwent yibum: Beis Shammai declare her fit, since in their view a mitzvah relationship took place here, the woman is not considered a zonah, and she is not disqualified from the kehunah. Beis Hillel disqualify her, because the rival wife of an ervah is exempt from yibum and from chalitzah, and since there is no mitzvah of yibum here, the prohibition of a brother's wife returns to its place - it turns out that a forbidden relationship took place here, and the woman is disqualified from the kehunah as a result.
The relationship between Beis Shammai and Beis Hillel:
The Mishnah teaches us something remarkable about the relationship between the two houses: "Af al pi she'eilu osrin v'eilu matirin, eilu poslin v'eilu machshirin - lo nimne'u Beis Shammai mi'lisa nashim mi'Beis Hillel, v'lo Beis Hillel mi'Beis Shammai" - even though these forbid and these permit, these disqualify and these declare fit, Beis Shammai did not refrain from marrying women from Beis Hillel, nor did Beis Hillel from Beis Shammai. Although in these cases one group forbids the woman and the other permits her, and one disqualifies her from the kehunah while the other declares her fit, Beis Shammai did not refrain from marrying women from the families of Beis Hillel, nor did Beis Hillel from the families of Beis Shammai.
Their practice was that they would inform one another wherever there was cause for concern. Beyond standing firmly by their own positions in these matters, they maintained harmonious relations with one another to the point of intermarrying, even though they had reservations about the fitness of certain individuals based on their conduct - and it was enough for them to notify each other where a problem might arise.
And more than that: "V'chol ha'taharos v'ha'tumos she'hayu eilu metaharin v'eilu metam'in - lo nimne'u osin taharos eilu im eilu" - and with all the matters of taharah and tumah in which these declared pure and these declared impure, they did not refrain from preparing items of taharah together with one another. Even in matters of tumah and taharah, where one house would declare a given item pure and the other would declare it impure, they did not hold themselves back from preparing items of taharah together: they cooked together, dealt together with matters of tumah and taharah, and performed other tasks together. They were careful only that nothing subject to dispute should arise between them, and they united around the things they agreed upon, while the matters in which they differed each kept to himself - and in this way they succeeded in maintaining a harmonious relationship with one another.
In summary: in this Mishnah we learned the position of Beis Shammai, who permit the rival wives to the brothers and dispute the principle exempting the rival wife of an ervah, along with the two practical consequences between the houses - in the case of chalitzah and in the case of yibum, regarding her fitness for the kehunah. We also learned the great principle at its conclusion: even where there was sharp disagreement in matters of prohibition and permission, of disqualification and fitness, and of tumah and taharah, Beis Shammai and Beis Hillel conducted themselves toward one another with open communication, with unity, and with intermarriage.