If the anointed priest sinned (in his ruling) and was afterwards deposed from his (high-) priesthood [and he was not able to bring his offering until after he had been deposed]; and, similarly, if the Nassi sinned and was afterwards deposed from his kingship — the anointed priest brings a bullock [(It is stated later in our Mishnah that even if he sinned after he had been deposed he brings a bullock. But because it was necessary to teach that the Nassi brings a he-goat before he was deposed, it was also taught that the high-priest brings a bullock], and the Nassi brings a he-goat, [it being written (Leviticus 4:28): "for his sin which he has sinned" — he brings a sin-offering as he did when he sinned.]
Bartenura on Mishnah Horayot
If an anointed priest transgressed and afterwards relinquished his priesthood - And he was unable to bring his offering until after he had already relinquished his priesthood.
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English Explanation of Mishnah Horayot
Introduction
In the case discussed by our mishnah, a high priest or king issued an errant ruling that was subsequently followed by the people, but before they could bring the appropriate sin-offering, they lost their office. The question is, are they still obligated to bring the same offering, even though they are no longer high priest or king.
As a background to this mishnah, I should point out that the removal of high priests and kings from their office was not an uncommon occurrence in the Second Temple period. Due to the political pressures of the Roman rulers, kings and especially high priests were often replaced.
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Bartenura on Mishnah Horayot
The anointed priest brings a bull offering - And even if he transgressed after he relinquished his priesthood, we say later on in our Mishna that he brings a bull offering. We only mention [the offering of an anointed priest] because we needed to teach that a ruler brings a he-goat when he transgresses before he relinquishes [his rulership], therefore we also teach [/mention] that an anointed priest brings a bull offering.
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English Explanation of Mishnah Horayot
If an anointed priest transgressed and afterwards relinquished his high priesthood, and similarly if a ruler transgressed and afterwards relinquished his reign, the anointed priest brings a bullock, and the ruler brings a he-goat. The mishnah rules that in the situation where the high priest or king issued an errant ruling but were removed from office before bringing their sin-offering, they still bring the same sin-offering that they would have brought had they still been in office. The obligation for the offering is set at the time of the errant ruling, and not at the time of the bringing of the sacrifice. Therefore, the high priest still brings a bull, and the king still brings a goat. In the next mishnah we will see that the high priest brings a bull even if he issued the errant ruling after he was removed from office. Therefore, our mishnah is not really necessary with regard to the high priest because he in any case brings a bull. The only reason that the mishnah teaches about the high priest is because it wants to teach about the ruler, who brings a goat only if he issues the errant ruling while still a king.
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Bartenura on Mishnah Horayot
And the ruler brings a he-goat. As it says (Leviticus 4:3) "for the sin that he transgressed," which teaches us that he brings a sin offering in accordance with his obligation at the time of his sin.