In the first Mishnah of the Perek (chapter), we learned that a firstborn animal that is partly owned by a non-Jew is exempt from the laws of Bechor (firstborn). This Mishnah talks about a tzon barzel arrangement. Tzon barzel means iron sheep. It was an arrangement usually done with sheep, where a person gives another individual an animal to raise for him, and the person raising it accepts full responsibility to return its value as it was, no matter what the situation is. In return, he gets to care for the animal and profit from it in some way. He receives a portion of the profits. The agreement says that, at a certain time, the full value of the item has to be returned to the owner. If the item is lost or destroyed, the person taking care of it agrees to pay back its full value. Tzon was usually done with sheep. Barzel means iron: just as iron is preserved and doesn't bend whatsoever, the original value of the item is preserved in perpetuity.
The Mishnah says: Somebody accepts an animal on contract from a non-Jew under a tzon barzel agreement. The offspring of the animal are exempt from the laws of Bechor. When the offspring grow up and give birth to a firstborn, that firstborn would not have the status of a Bechor, even though its mother primarily belongs to a Jew. The idol worshiper has the right to seize it if the Jew doesn't follow the obligations under the tzon barzel agreement. So the idol worshiper, or the non-Jew, is considered to have some ownership in the mother. Therefore, it exempts the firstborn from the laws of Bechor. We're even talking about the portions of the profits of the Jew. Let's say the Jew gets the first Bechor. He's entitled to it fully. We still say that he's exempt in this case because of the tzon barzel arrangement between the Nochri, the non-Jew, and the Jew.
But the offspring of the offspring of the original tzon barzel animals, the second generation of tzon barzel animals, are subject to the laws of Bechor. Their firstborn, which is the third generation, has the status of Bechor. Once the idol worshiper gives up the profits and everything is distributed, that becomes the full property of the Jew. Therefore, the non-Jew has no ownership in the second generation. If the second generation gives birth to a Bechor, those firstborn animals are subject to the laws of Bechor.
Now we'll bring down a situation where even the offspring of the second generation are not a Bechor. If the Jew set up the offspring of the original tzon barzel animals in place of their mother, he specified that the idol worshiper is allowed to seize the first generation of offspring. Mothers aren't around anymore. It's like a little bit of a long-term setup just to guarantee to the original owner that he'll get something back. Then even the offspring of the original animal's offspring, which is the third generation and would be a Bechor, are exempt from the laws of Bechor, even if the firstborn of the second generation—which becomes the third generation—even though it wasn't part of the original agreement. By pledging that the first generation of offspring could serve as a replacement for the mother if the mother dies, the Jew now gave over some rights to the non-Jew even in the second generation of offspring, and he allows them to be taken. Therefore, he created a situation where the non-Jew has some sort of ownership all the way through a later generation, and there would be no Bechor in that case as well. So the firstborn of the second generation of offspring would be exempt from the laws of Bechor. But in this case, the offspring of the offspring's offspring, which is the third generation of offspring, would be subject to the laws of Bechor. The firstborn of these offspring, which is the fourth generation, is considered to be a Bechor. The idol's rights are expanded to include only one additional generation of offspring, which is the second generation, but not the third generation. Therefore, he has no ownership in that third generation of animals, and the firstborn of those animals could have the status of a Bechor.
Says, even if the Jew set up the offspring of the original tzon barzel animals in place of their mothers, then even as many as 10 generations of animals are exempt from the laws of Bechor. It goes all the way down. Why? Because they are committed to the idol worshiper. The idol worshiper has the right to seize these animals if the Jew doesn't pay for the original tzon barzel animals. According to Rashbag, when a Jew pledges the original offspring to replace the mother, he's not only including them but also all the future generations. It goes in perpetuity, no matter how far removed they are from the original animals. Therefore, even those offspring would be exempt from the laws of Bechor.