Eruvin, Chapter 3, Mishnah 8. In the previous mishnah we learned that Rabbi Yehudah holds that the two days of Rosh Hashanah are a case of doubt - a doubt as to on which of the two days Rosh Hashanah actually falls - and therefore one may place an eruv in two different directions; the Sages disagreed and ruled that both days of Rosh Hashanah have the status of one long single sanctity. The basis of this dispute touches on additional areas of halachah, and our mishnah deals with them.
Separating Terumah and Maasros on Yom Tov:
One does not separate Terumah and maasros on Shabbos or on Yom Tov. However, according to Rabbi Yehudah's understanding of Rosh Hashanah, that the matter is in doubt and only one of the two days is truly Yom Tov, a person may separate maaser on condition, so that the separation will take effect on one of the two days.
How so? A person makes a stipulation regarding his basket of produce:
On the first day: "If today is Yom Tov - I have no power to separate Terumah and maasros; and if today is Erev Yom Tov and tomorrow is Yom Tov - it is my wish that the maasros take effect today."
On the second day: "If today is Yom Tov - I have no power to separate Terumah and maasros, which means that yesterday's separation took effect and I may eat from the produce today; and if today is not Yom Tov, so that yesterday's separation did not take effect - it is my wish that today's separation take effect."
The result is that however one looks at it, by the second day Terumah and maasros have already been separated, and therefore he is permitted to eat from the produce on the second day.
An Egg Laid on the First Day:
The law is similar with an egg laid on Yom Tov, which may not be eaten until after Yom Tov. But an egg laid on the first day of Rosh Hashanah is permitted on the second day of Rosh Hashanah, since it is permitted either way:
If the first day was Rosh Hashanah - then the second day is not Rosh Hashanah but the day after it, and the egg is certainly permitted.
And if the first day was Erev Rosh Hashanah and the second day is Rosh Hashanah - then the egg was laid on Erev Yom Tov and not on Yom Tov, and in that case too it is permitted.
The View of the Sages:
The Sages did not agree with Rabbi Yehudah, because they viewed Rosh Hashanah as one long sanctity, with everything that occurs within it treated accordingly - as though the first day of Rosh Hashanah were the first half of a single day. Therefore one may not make such a stipulation regarding the separation of maaser, and likewise an egg laid on the first day remains forbidden even on the second day.
It is worth noting that the Sages stated their ruling with regard to Rosh Hashanah alone; but regarding the second day of Yom Tov observed in the Diaspora, on the other festivals, they agree with Rabbi Yehudah that these are two separate days - and therefore one may make the stipulation regarding the separation of maaser, and the egg is permitted on the second day.
In summary: This mishnah continues the basis of the dispute we saw in the previous mishnah. According to Rabbi Yehudah, the two days of Rosh Hashanah are a case of doubt, and therefore one may make a stipulation in separating Terumah and maasros that it take effect on one of the two days, and likewise an egg laid on the first day is permitted on the second day. According to the Sages, the two days are one long sanctity, and therefore a stipulation regarding the separation of maaser is of no avail and the egg remains forbidden - and all this applies to Rosh Hashanah alone, but regarding the second day of Yom Tov observed in the Diaspora, they too agree with Rabbi Yehudah.