Gemara

113b (Rav Shmuel bar Rav Yitzchak): "V'Yorash" equates the second inheritance to the first inheritance. Just like sons inherit before daughters in the first inheritance, also in the second.

115a (Mishnah): The order of inheritance: "Ish Ki Yamus u'Ven Ein Lo v'Ha'avarta Es Nachalso l'Vito" - a son precedes a daughter, and all the descendants of a son precede a daughter;

A daughter precedes the brother of the deceased, and all her descendants precede a brother;

A brother precedes the brothers of the father, and all his descendants precede the father's brothers;

The general rule is, whoever has precedence to inherit, all his descendants have precedence.

Rishonim

The Rif and Rosh (8:7) bring the Mishnah.

Rambam (Hilchos Nachalos 1:3): Whoever has precedence to inherit, all his descendants have precedence. If a man or woman (Simchah) died and left a son, he inherits everything. If the son died, we see if there are any male or female descendants of the son, any number of generations down. Even if it is his daughter's daughter's daughter, she inherits everything. If not, we go to the daughter. If she is alive, she inherits everything. If she died, if she left any descendants, he or she inherits everything. If not, the inheritance reverts to the father. If he died, we see if he has descendants, i.e. Simchah's brothers. Males and their descendants have precedence over females, just like regarding Simchah's descendants. If he has a brother or a descendant of a brother, he inherits everything. If not, we see if Simchah left a sister or descendants of a sister. If there is, she inherits everything. If not, the inheritance reverts to the grandfather.

Magid Mishneh: "V'Yorash" equates the second inheritance to the first inheritance. Just like sons inherit before daughters in the first inheritance, i.e. of the deceased, also in the second, i.e. of his relatives.

Rambam (5): If one had two sons, and they died in his lifetime, and one left three sons and the other son left a daughter, the three grandsons inherit half, and the granddaughter inherits half. Each inherits what was due to his father. The same applies to nephews or cousins.

Rambam (2:7): If one had two sons, and one was a Bechor, and they died in his lifetime, and the Bechor left a daughter and the other son left a son, the grandson inherits a third, which was due to his father, and the granddaughter inherits two thirds, which was due to her father. The same applies to nephews or cousins. If the father of an heir was a Bechor, the heir inherits his extra share.

Objection (Tur CM 277:24): I do not understand this. Bechorah applies only to a son inheriting a father, but not to other heirs!

Magid Mishneh: If Yakov died in the life of his father Yitzchak, and after Yitzchak died later, Yakov's Bechor Reuven does not get a double share in Yitzchak's property, for it was only Ra'uy to fall to Yakov. However, if Yakov was a Bechor, his children inherit his double share of Yitzchak's property. We cannot say that the Rambam discusses when Yakov died without children, and Yitzchak was dead and Yitzchak's descendants inherit. A Bechor would not get double, for Yakov's property was Ra'uy to come to Yitzchak! Rather, Yakov died, and afterwards Yitzchak died. However, why did he need to discuss a case like this? It does not matter how the property came to Yitzchak. His Bechor, or descendants of his Bechor, receive a double portion in any case!

Bach (CM 277:24): The Rambam's intent is clear, like he wrote in (1:5). He explains 'the second inheritance is like the first inheritance' like Tosfos, that the first inheritance is what is explicit in the Torah, and the second inheritance is grandchildren. The Hekesh equates them (the children of explicit heirs inherit like their parents would inherit). The same applies to Bechorah. If all Yakov's children died, the heirs of his Bechor inherit the double portion due to their father. This applies to any heirs of Yakov's children, e.g. nephews (of Yakov's grandchildren). Bechorah applies to anyone who inherits Yakov's Bechor. The Semag copied the Rambam without challenging it, for it is clear like I explained.

Poskim

Shulchan Aruch (CM 276:1): The order of inheritance: if Simchah died, his son inherits. If the son died, if there are any male or female descendants of the son, any number of generations down, he stands in place of the son and inherits everything. If not, if there is a daughter, she inherits everything. If she died, if she left any descendants, he or she inherits everything. If not, the inheritance reverts to the father. If he died, if he has a descendant, i.e. Simchah's brother, he inherits everything. If he has no brother, but there is a descendant of a brother, he inherits everything. If not, we see if Simchah left a sister or descendants of a sister. If there is, she inherits everything. If not, the inheritance reverts to the grandfather.

Chasam Sofer (EH 2:158, cited in Pischei Teshuvah 1): A case occurred in which a man wouned his mother and made her bleed. The congregation fined him, and forced him to say 'I remove myself from inheriting her when she will die.' He signed on this. His mother died, and he claims that one cannot remove himself from Torah inheritance. This is true, like the Ran and Beis Shmuel (Sof Siman 92) say, unlike Chavos Ya'ir and Nachalas Shiv'ah. However, here it was a fine. We force him to fulfill his vow not to benefit from the property. A husband must pay a fine that his wife owes, for it is like a Korban, which he must bring for her. The Rambam and Shulchan Aruch (CM 212:7), who rule that if one said 'I will give to Aniyim what will come for this tree', even though it is Davar she'Lo Ba l'Olam, it is a vow and we force him to fulfill it. Surely, the same applies here. The Rema (EH 91) favors those who say that it is not a vow (for he did not forbid anything), and he need not fulfill it. Here, he accepted not to receive the money, to atone for his sin. It is like a vow not to eat. However, if he has sons, they can inherit their grandmother. Our fine is no stronger than the Torah's law regarding one who stole from his father, according to Tosfos (Bava Kama 108b DH l'Vanav. He must remove the theft from his Reshus, but his children may take it.)

Shulchan Aruch (4): The mother's family is not considered family. A woman does not inherit her son or daughter. Maternal brothers from different fathers do not inherit each other. Rather, paternal heirs inherit.

Rema: Maternal relatives do not inherit even a Shtuki (one who does not know whom is his father. His property is Hefker after he dies), even though he has no paternal heirs.

Pischei Teshuvah (2): A case occurred in which a man came to a city, married Leah, fathered a girl Dinah, divorced Leah and went away. No one knows about the man. Leah died, leaving a house rented out to Ploni. Surely, Leah's family has no claim to inherit her. The question is whether Dinah's property is Hefker, like that of a Shtuki. Mayim Chayim (CM 24) answered that this is unlike a Shtuki. There, the father has no Chazakah, even if Eliyahu would tell us that he is the father. Even though inheritance does not need testimony, it needs a Chazakah that one was a relative (CM 284). Here, the father was known, and he could come and clarify this. This is like one who saw an Aveidah fall from two or three people, but does not know from whom. He holds it until Eliyahu comes (CM 262:4).

Chavos Ya'ir (231, b'Sof): If one seized property of a Shtuki, he keeps it, for it is Hefker, like the Rema says. This requires investigation. It seems that he should leave it until Eliyahu comes.