Bechorot 58 - Three Dates for Animal Tithes
There are three dates designated for animals tithes: fifteen days before the holidays of Passover, Shavuot, and Sukkot. Thirty days before a holiday one starts reviewing its laws, and these dates were in the middle. After the designated date, the animals are considered "untithed." Although there are opinions that the tithing calendar is somewhat different, all agree that these dates were established by the earlier Sages in order that enough animals were available for those going to Jerusalem and bringing sacrifices on the holidays.
What is the New Year for animal tithe? Rabbi Meir says that it is the first of Elul, a month before the Rosh HaShana of the first of the month of Tishrei, but Rabbi Elazar and Rabbi Shimon both say that it is the same as Rosh HaShana. They argue about when the majority of births occur. The practical importance of this day is that only animals born in the same year combine for the count of ten needed to create the tithe obligation. Ben Azzai says that the New Year for tithe is a very old disagreement which cannot be properly resolved, and therefore animals born in Elul form a separate tithing queue.
Art: Theophile Louis Deyrolle - A Shepherdess and her Flock