Tractate Bechoros, chapter 9, Mishnah 6. The Mishnah deals with another practical matter: how to calculate animal tithes.
The Mishnah states: "Kol shenoldu me'echad b'Tishrei ve'ad esrim vetishah b'Elul - harei elu mitztarfin" - any animal born from the first of Tishrei until the last day of the month of Elul, meaning within the calendar year, combines into one tally. If the flock owner had ten or more animals born that year, he must tithe them; if less than that, he does not tithe.
"Chamishah lifnei Rosh Hashanah vechamishah le'achar Rosh Hashanah - einan mitztarfin" - if he had in his flock five lambs born one day before Rosh Hashanah and another five born one day after it, and these were all the lambs born to him over the two years, they do not combine, because only five were born in each year, even though they were all born within a span of two days. Therefore, he does not separate a tithe at all.
The three gathering times:
In the previous Mishnah we learned that by rabbinic law, three cutoff dates were established when one must tithe, three times during the year, and their time is fifteen days before each of the festivals. Therefore, the last day of Adar, fifteen days before Pesach, is a designated time (a 'goren').
"Chamishah lifnei hagoren vechamishah le'achar hagoren - harei elu mitztarfin" - if five lambs were born on the last day of Adar, and another five the next day on the first of Nisan, they combine with each other, and when the second goren arrives, fifteen days before Shavuos, he will combine them together into one tally.
"Im ken lamah ne'emar shalosh granos lema'asar behemah?" - what is the significance of these designated times, if they do not serve as a cutoff point for the calculation?
"Ad shelo higi'a hagoren - mutar limkor velishchot" - as long as the designated time has not arrived, and the obligation to tithe is not imposed on him, the flock owner is permitted during the interim period to slaughter the animals, eat them, and sell them, and they do not enter the tithing calculation at all.
"Higi'a hagoren - lo yishchot" - once the time has arrived, all the animals born in that calendar year up to that time are examined; they are brought into the pen, and one out of every ten becomes the animal tithe. From here on it is forbidden to slaughter or even sell them.
"Ve'im shachat - patur" - one who slaughters and eats the animals after the designated time only violates a rabbinic prohibition, and therefore he is exempt; by Torah law he was permitted to wait the entire year and tithe them all together.
A practical example of the calculation:
The first goren: Fifteen lambs were born to a person that year, and the last day of Adar arrived, fifteen days before Pesach. He brings them into the pen, and the tenth one to emerge becomes the animal tithe. The remaining five are not tithed for the time being.
The second goren: During the next two months another fifteen were born to him, making a total of thirty animals that year. When the second goren arrives, fifteen days before Shavuos, on the thirty-fifth day of the Omer, he combines the fifteen new ones with the five remaining from the previous time. Twenty animals are brought into the pen, and two emerge from them - the tenth and the twentieth - and they are the animal tithes.
And if during the interim period, between Pesach and Shavuos, he chose to eat one of the lambs, he is permitted to do so. Then he is left with fourteen from this period and five from the previous period, a total of nineteen animals. They are all brought into the pen, only one emerges as the tenth, and the remaining nine are left over and are not tithed now, but rather are kept for the third period, and so on.
In summary: In this Mishnah we learned that combining animals for tithing is determined by the calendar year, from the first of Tishrei until the twenty-ninth of Elul, and those born before Rosh Hashanah do not combine with those born after it. The three cutoff dates established by rabbinic law do not separate between the animals - those born before the goren and after it combine - and their purpose is to establish from when it is forbidden to sell and slaughter, and therefore the remaining ones that did not reach a total of ten are left over and combine for the next goren.