Talmud for Sanhedrin 2:1
כֹּהֵן גָּדוֹל דָּן וְדָנִין אוֹתוֹ, מֵעִיד וּמְעִידִין אוֹתוֹ, חוֹלֵץ וְחוֹלְצִין לְאִשְׁתּוֹ, וּמְיַבְּמִין אֶת אִשְׁתּוֹ, אֲבָל הוּא אֵינוֹ מְיַבֵּם, מִפְּנֵי שֶׁהוּא אָסוּר בָּאַלְמָנָה. מֵת לוֹ מֵת, אֵינוֹ יוֹצֵא אַחַר הַמִּטָּה, אֶלָּא הֵן נִכְסִין וְהוּא נִגְלֶה, הֵן נִגְלִין וְהוּא נִכְסֶה, וְיוֹצֵא עִמָּהֶן עַד פֶּתַח הָעִיר, דִּבְרֵי רַבִּי מֵאִיר. רַבִּי יְהוּדָה אוֹמֵר, אֵינוֹ יוֹצֵא מִן הַמִּקְדָּשׁ, שֶׁנֶּאֱמַר (ויקרא כא) וּמִן הַמִּקְדָּשׁ לֹא יֵצֵא. וּכְשֶׁהוּא מְנַחֵם אֲחֵרִים, דֶּרֶךְ כָּל הָעָם עוֹבְרִין בָּזֶה אַחַר זֶה וְהַמְמֻנֶּה מְמַצְּעוֹ בֵּינוֹ לְבֵין הָעָם. וּכְשֶׁהוּא מִתְנַחֵם מֵאֲחֵרִים, כָּל הָעָם אוֹמְרִים לוֹ אָנוּ כַפָּרָתְךָ, וְהוּא אוֹמֵר לָהֶן תִּתְבָּרְכוּ מִן הַשָּׁמָיִם. וּכְשֶׁמַּבְרִין אוֹתוֹ, כָּל הָעָם מְסֻבִּין עַל הָאָרֶץ וְהוּא מֵסֵב עַל הַסַּפְסָל:
A high-priest judges and he is judged; he testifies and he is testified for; he administers chalitzah and his wife receives chalitzah. His wife is taken in yibum, but he does not perform yibum, for a widow is forbidden to him. If one of his kin died, he does not follow the litter, [lest he come to touch it in his preoccupation, and it is written (Leviticus 21:11): "And to any dead body he shall not come."] But they (the litter bearers) are concealed and he is revealed; they are revealed and he is concealed. [When the litter bearers are "concealed" from the alleyway that they left, he is "revealed," and enters it. But so long as they are "revealed" in the alleyway, he is "concealed," from it, not entering it.] And he goes out with them until the entrance of the city. [For there are alleyways in the city in which he can conceal himself, but outside the city there is no (distinguishing) sign (of his separation from the procession)]. These are the words of R. Meir. R. Yehudah says: He does not go out of the sanctuary, viz. (Leviticus 21:12): "And from the sanctuary he shall not go out." [R. Yehudah expounds: "And from the sanctuary (mikdash), he shall not go out" — at all. R. Meir expounds: "And from his sanctity (kedushatho), he shall not go out," i.e., He should take care to avoid contact (with the body). And within the city, where there are alleyways, there is a distinguishing) sign, so that he takes care. The halachah is in accordance with R. Yehudah.] And when he consoles others, (when) it is the custom of all the people to pass by one after the other [(For all agree that he may follow the litter of one who is not his kin; for not being preoccupied, he will take care to avoid contact with the litter. And when they return from burial and stand in a file to console the mourners, who stand in their halts, and each one says: "Be consoled by Heaven")], the memunah (the "appointed one") [the adjutant high-priest, who is appointed to substitute for the high-priest if he becomes unfit on Yom Kippur] positions him in the middle between himself and the people. [For the memunah walks to the right of the high-priest, and all the people to his left, so that the high-priest is found to be in the middle.] And when he is consoled by others, all the people say to him: "We are your atonement" [i.e., You will receive atonement through us, and we are in your place for all that should come upon you], and he says to them: "Be blessed by Heaven!" And when they give him the mourner's meal, [for a mourner is forbidden to eat the first meal of his own food, and his kin and friends provide it for him], everyone sits on the ground [constricting themselves, and mourning for his suffering], and he sits [in (regard for his) dignity] on a bench.
Jerusalem Talmud Horayot
An Anointed Priest who was removed from his anointed status and then sinned, or a Prince who was removed from his exalted status and then sinned: The Anointed Priest brings a bull and the Prince is like a commoner2Again the moment of the sin determines the kind of sacrifice. Priestly offices follow the rule that “one rises in holiness; one does not descend;” the former Anointed Priest remains in his status. He cannot officiate as Anointed Priest since another one holds the office; neither can he officiate as common priest since he may not descend in rank. The same is true for an unanointed High Priest. But a politician who loses his rank is free to run for any other office; his sacrifice is a female goat or sheep like everybody else’s..