Yoma, Chapter 7, Mishnah 3. The Mishnah continues from the point at which the Kohen Gadol finished reading from the Torah, and details the order of sanctifying the hands and feet, the changes of garments and the immersion that follow the reading, and the order of the offerings performed in the golden garments.
He read from the Torah in the white garments:
If he read from the Torah while still wearing the white garments - the four garments in which he performed the service that preceded the reading - and did not change out of them, this is the order of what he does:
He sanctifies his hands and feet.
He removes the white garments.
He goes down and immerses, comes up and dries himself.
They bring him the golden garments - the eight garments he wears throughout the year - and he puts them on.
He again sanctifies his hands and feet.
Why a sanctification of the hands and feet is required here:
The Mishnah is precise: specifically one who read from the Torah in the white garments is required to sanctify his hands and feet before removing them, since he has not yet removed them. But according to the first Mishnah in this chapter, the Kohen Gadol may read from the Torah in his own white robe - in his own white tunic, which is not one of the priestly garments. In such a case he already sanctified his hands and feet and removed the white garments when he changed into his own robe, and therefore there is no need to sanctify his hands and feet a second time before the immersion.
The view of Rabbi Eliezer:
After he has put on the golden garments, the Mishnah continues: he goes out and performs "eilo v'eil ha'am" - the ram he is commanded to bring on Yom Kippur and the ram brought on behalf of the Jewish people - "v'shivas kevasim temimim benei shanah" - and seven unblemished lambs in their first year. This is the view of Rabbi Eliezer.
The Gemara explains this view on the basis of a beraisa, that this going out takes place at the third immersion:
The first immersion: in the golden garments, for the service of the morning tamid.
The second immersion: in the white garments, for the services performed in the Sanctuary and in the Holy of Holies.
The third immersion: in the golden garments, at which he performs his ram and the ram of the people, and also burns the emurim of the bull and of the goat.
The seven lambs, however, which are part of the Musaf offerings, are not brought at the third immersion. They, together with the remaining components of the Musaf - the bull brought as an olah and the goat brought as a chatas - are offered with the afternoon tamid, at the final immersion, the fifth immersion. This is how the Gemara explains this line in the Mishnah.
The view of Rabbi Akiva:
Rabbi Akiva disagrees and holds that the seven lambs were offered at the first immersion, together with the morning tamid, and along with them the bull brought as an olah, which is also part of the Musaf offerings.
And in the closing words of his statement: "v'hasa'ir hana'aseh bachutz" - and the goat that is performed outside - this does not refer to the earlier goat that was brought into the Sanctuary and the Holy of Holies, but to the goat that is part of the Musaf offerings, "na'aseh im tamid shel bein ha'arbayim" - it is performed together with the afternoon tamid.
The meaning is not that it is performed literally together with the afternoon tamid, but rather that it is performed later than the other parts of the Musaf - the seven lambs and the bull brought as an olah, which were offered with the morning tamid - and closer to the afternoon tamid. That is, at the third immersion, at which he performs his ram and the ram of the people and the emurim of the bull and the goat. This goat was separated from the rest of the Musaf offerings, but it was not actually offered together with the afternoon tamid; all that is novel about it is that it is not performed with the morning tamid, as the seven lambs and the bull brought as an olah are.
In summary: in this Mishnah we learned the order of the change of garments following the Torah reading - sanctifying the hands and feet, removing the white garments, immersing and drying, putting on the golden garments and sanctifying again - and the reason this sanctification is required specifically of one who read in the white garments and not of one who read in his own robe. We also examined the dispute between Rabbi Eliezer and Rabbi Akiva regarding the time the Musaf offerings are brought: according to Rabbi Eliezer the seven lambs, the bull brought as an olah and the goat brought as a chatas are offered with the afternoon tamid at the fifth immersion, while according to Rabbi Akiva the seven lambs and the bull brought as an olah are offered with the morning tamid, and the goat that is performed outside is offered at the third immersion.
We will continue with the fifth immersion and the services performed together with it, with the afternoon tamid.