Gemara

(If Reuven gives animals to Shimon to be "Tzon Barzel," we assess their value. Shimon has a set time to pay, and until then Reuven gets half of the offspring. We will call the animals given "mothers.")

(Mishnah): If Shimon received Tzon Barzel from a Nochri, the Vlados (offspring) of the mothers are exempt from Bechorah. Vladei Vlados (grandchildren) are liable.

If he designated children in place of their mothers, Grandchildren are exempt, and great grandchildren are liable.

R. Shimon ben Gamliel says, even 10 generations are exempt, for the Nochri has Acharayus. (He may collect from them.)

Inference: Because the owner did not receive money for the animals, it is as if they still belong to the original owner (the Nochri. Therefore, they are exempt.)

Contradiction (Mishnah #1): Reuven may not receive Tzon Barzel from Shimon, for it is Ribis. (Shimon, the original owner, receives half of the offspring because he lent animals.)

(This would not be Ribis if the animals still belonged to Shimon. Rather,) this shows that the animals are considered to belong to the receiver!

Answer #2 (Rava): In both Mishnayos, the investor did not accept Ones or Zol;

(In our Mishnah), they are exempt from Bechorah because if the Nochri demands his money and the Yisrael does not give, the Nochri will take the animals. If he needs, he will take the children.

Whenever a Nochri has rights to collect from animals, they are exempt from Bechorah.

(Mishnah): If he designated children in place of their mother, Grandchildren are exempt.

(Rav Huna): (In the Reisha, without designation,) children (of the mothers) are exempt from Bechorah (a firstborn male is not Kadosh, nor the firstborn of a female Vlad), but a firstborn of grandchildren gets Kedushas Bechor.

(Rav Yehudah): Grandchildren are exempt, but a firstborn of great grandchildren is Kodesh.

Question (against Rav Yehudah - Mishnah): If he designated children in place of their mother, grandchildren are exempt.

Inference: Had he not designated them, grandchildren would be liable!

Answer: No, even had he not designated them, grandchildren would be exempt;

The Mishnah teaches that designation does not change anything. With or without designation, the Nochri normally collects from the grandchildren if he needs to. Therefore, grandchildren are exempt, but great grandchildren are liable.

Rishonim

Rambam (Hilchos Bechoros 4:4): If one accepts Tzon Barzel from a Nochri for a fixed price, and stipulated that they will divide the profits, and if there is a loss, the Yisrael will suffer it, even though they are in the Yisrael's Reshus and they are like his acquisition, since if the Nochri will not find other money to collect from, he will collect from these animals and their offspring, it is as if he has Acharayus (a lien) on them and their offspring. Since he has a hand in them, they and their offspring are exempt from Bechorah. The grandchildren are obligated, since they are the Yisrael's and the Nochri has no share in them.

Ri Korkus: The Rambam rules like Rav Huna. Why did the Rambam omit the case in which he designated children in place of their mothers? Perhaps because he holds that this means that the children become like (new) Tzon Barzel, he already gave the law. (Children of the (new) Tzon Barzel are exempt, and the grandchildren are obligated.) Perush ha'Mishnayos says that Vlados (children) are obligated. It seems that this is a textual error. However, I saw an accurate text (of Perush ha'Mishnayos) that says like our texts. Perhaps 'Vlados' refers to the great grandchildren; the Rambam merely teaches that the Halachah does not follow R. Shimon ben Gamliel, who exempts even 10 generations.

Kesef Mishneh: The Rambam holds that Rav Huna and Rav Yehudah argue only about for how many generations the Nochri normally has Acharayus on the offspring. The general rule is that any offspring on which the Nochri has a lien are exempt. Offspring on which he has no lien are liable. This is why the Rambam omitted the case in which he designated children in place of their mothers.

Lechem Mishneh: The Gemara answered that Rav Yehudah holds that designation of children in place of their mothers has no effect. The Tur wrote that Rav Huna and Rav Yehudah argue about designation. Rav Huna holds that this adds a generation, and Rav Yehudah holds that it does not. Really, their argument is about without designation! Why didn't the Rambam teach about designation? The Kesef Mishneh says that the Rambam's rule teaches the law; both Amora'im agree to it. This is wrong. The Rambam rules like Rav Huna, for he said Stam that children are exempt and grandchildren are obligated, like Rav Huna. Rather, the Rambam explains designation like Tosfos, that the mothers died, and he set the children in place of them. They become Tzon Barzel. Rav Huna holds that the Chidush is the inference, that without designation the grandchildren are liable, unlike Rav Yehudah. The Rambam taught Rav Huna's law, therefore he did not need to teach about designation.

Lechem Mishneh: The Tur explains that the Rambam is unlike Rashi. The Rambam exempts even if the Nochri receives only money of the profits. Rashi exempts only if the Nochri has a share in the children themselves.

Rosh (2:2): Rav Huna holds that the fourth generation from the Tzon Barzel are given to a Kohen. Rav Yehudah holds that the fifth generation is given. The Ramban says that the Gemara did not say whom the Halachah follows, therefore the fourth generation is a Safek. It seems to me that the Halachah follows Rav Huna, for he was greater than Rav Yehudah. In several places, Shmuel, the Rebbi of Rav Yehudah, asked questions from Rav Huna. Also, we challenged Rav Yehudah from the Mishnah, and his answer was weak. It seems that R. Yonah holds like this.

Poskim

Shulchan Aruch (YD 320:7): If one accepts Tzon Barzel from a Nochri for a fixed price, and stipulated that they will divide the profits, and if there is a loss, the Yisrael will suffer it, even though they are in the Yisrael's Reshus and they are like his acquisition, since if the Nochri will not find other money to collect from, he will collect from these animals and their offspring, it is as if he has Acharayus on them and their offspring. Since he has a hand in them, they and their offspring are exempt from Bechorah. The grandchildren are obligated, since they are the Yisrael's and the Nochri has no share in them. If he designated children in place of their mothers, grandchildren are exempt, and great grandchildren are liable.

Beis Yosef (DH u'Mah she'Chasav sheha'Rosh): The Rosh and Rambam rule like Rav Huna.

Bach (7): The Rambam did not rule like Rav Huna, and not like Rav Yehudah. He did not count generations. He said that the children are exempt and the grandchildren are liable. He connotes that all agree to this, since the Nochri has no share in the children. I do not know why the Tur says that the Rambam rules like Rav Huna. Perhaps he had another text of the Rambam.

Beis Yosef (DH u'Mah she'Chasav v'Im): Rashi explains that 'designating children in place of their mothers' means that he offered the Nochri rights to collect from the offspring. He said 'if the (initial) animals will die, you will collect from the children that I should have received. They will be under my hand for that amount until the time itself.' This gives the Nochri power to collect from an extra generation. Therefore, the children of those children, which are grandchildren of the Tzon Barzel, are exempt, for the Nochri has a hand in them. When they have a Bechor, it is not given to a Kohen.