Yoma, Chapter 6, Mishnah 6. This mishnah describes the closing stages of sending the goat to Azazel: what the appointed man did when he reached the cliff, at the edge of the abyss.
The wording of the mishnah: "Mah hayah oseh? Cholek chut shel shani - chetzyo kashar ba'sela va'chetzyo kashar bein shtei karnav, u'dchafo la'achorav, v'hu misgalgel v'yored" - What did he do? He divided a strand of crimson wool - half of it he tied to the rock and half of it he tied between the goat's two horns, and he pushed it backwards, and it went rolling down. The goat would not even reach halfway down the mountain before it became "eivarim eivarim" - limb by limb, completely shattered into pieces.
Why the crimson strand was divided in two:
The Gemara explains: the crimson strand would turn white as a sign that the sins of Israel had been atoned for and forgiven. Therefore they tied part of it to the rock, so that it would be clearly visible that the strand had turned white. Had all of it remained between the goat's horns, the goat might have landed in a position where the strand could not be seen, and the matter would have remained in doubt.
Why it was not all tied to the rock:
The concern was that the strand might turn white immediately, before he pushed the goat off the cliff. Out of joy, and out of his sense that the atonement had already taken place, he might think that there was no longer any need to complete the mitzvah and push the goat, since it had already turned white. Therefore they divided the strand in two: even if the half on the rock turned white, since at that moment the man is occupied with tying the other half to the goat's horns, he continues with the process he has already begun and pushes the goat down the mountainside.
The return to the last booth:
Once he finished his task, the mishnah says: "U'va v'yashav lo tachas sukkah acharonah ad she'techshach" - and he came and sat under the last booth until nightfall - until Yom Kippur had ended. Now, ordinarily one who has gone outside the Shabbos boundary, even with the permission of the Sages as in this case, has only two thousand amos in every direction, whereas here he is four thousand amos - two mil - away from the last booth. But because of the danger of remaining in the wilderness alone and unprotected until nightfall, they permitted him to return to that booth.
From what point does he become tamei:
The Torah establishes that the one who sends the goat to Azazel becomes tamei, and not only does he himself require immersion in a mikvah, but his garments and the things he touches also become tamei. The question is at what stage this tumah takes effect, rendering his clothing and garments tamei. On this there is a dispute:
The view of the mishnah: from the moment he left the walls of Jerusalem - from that moment on he is already tamei.
Rabbi Shimon: he becomes tamei only from the moment he pushes the goat off the cliff; that is the point at which he becomes tamei.
In summary: in this mishnah we learned the closing stages of sending the goat - the division of the crimson strand between the rock and the goat's horns, and the reasons for it: so that the sign of the strand turning white would be visible to the eye, and on the other hand so that its turning white would not prevent the man from completing the mitzvah. We also learned about the permission for him to return to the last booth because of the danger, even though he is two mil away, and the dispute between the mishnah and Rabbi Shimon as to from when he renders garments tamei - from the moment he left the wall of Jerusalem or from the moment he pushed the goat off the cliff.