Gemara

(Mishnah): Semichah (pressing on the Korban's neck) applies to all Korbanos Yachid, except for Bechor, Ma'aser and Pesach.

92b: It says 'Korbano' three times. One of them teaches that Reuven may not be Somech on Shimon's Korban.

Gitin 28a - Question: Why does our Mishnah say that if one sent a Chatas from abroad, we (assume that he is still alive, and) offer it? The owner of a Chatas must come himself to do Semichah!

Answer #1 (Rav Yosef): The case is, a woman sent it. (Women do not do Semichah.)

Answer #2 (Rav Papa): The case is, he sent Chatas ha'Of. (Semichah does not apply to birds.)

Pesachim 62a: An Arel or Tamei can send any Korban through a Shali'ach (agent), except for Pesach (for it must be eaten).

Yevamos 64b (Beraisa #1 - Rebbi): If a woman circumcised two of her sons and they died, she should not circumcise her third son;

R. Shimon ben Gamliel says, she circumcises the third, but not a fourth.

Question: Another Beraisa (#2) switches the opinions. Which was taught later?

Answer (R. Chiya bar Aba): There was a case of four sisters in Tzipori. Three circumcised their sons, and all died. The fourth came to R. Shimon ben Gamliel. He told her not to circumcise her son. (This supports Beraisa #1; it was taught later.)

Objection: Perhaps he would have told also the third not to circumcise! R. Chiya's Chidush is that a Chazakah may be established through sisters.

(Rava): Since sisters make a Chazakah, a man should not marry a woman from a family of epileptics or lepers, i.e. if there are three in the family.

Rav Yosef brei d'Rava: Rav Yosef told me that the Halachah follows Rebbi. Another time, he told me that the Halachah follows R. Shimon ben Gamliel!

Rava: Regarding marriage and lashes, the Stam Mishnah is like Rebbi. Regarding Vestos (expected menstrual periods) and an ox Mu'ad (established) to gore, the Stam Mishnah is like R. Shimon ben Gamliel.

66a: Only one who may eat Terumah permits others to eat.

Objection: An Arel or Tamei Kohen may not eat, but he causes others to eat!

Answer: (An Arel or Tamei Kohen may eat,) just his mouth hurts him.

Shabbos 134a - Abaye's (surrogate) mother: If a baby is red, his blood has not been absorbed. If he is yellow, he does not have enough blood yet. In both cases we delay Milah until his blood is absorbed/he has enough blood.

(Beraisa - R. Noson): I met two woman who had lost two sons through circumcision. One's third baby was red, and the other was yellow. I counseled her to postpone the Milah until the blood was absorbed/the blood comes in. They did so, and the baby survived.

Rishonim

Rif and Rosh (Yevamos 20a and 6:13): Presumably, Milah is like marriage, because we are lenient about mortal danger.

Rambam (Hilchos Milah 1:18): If a woman circumcised her first son, and he died due to Bris, i.e. it weakened his strength, and similarly with her second son, whether he was from the same or a different father as the first, she does not circumcise the third in his time. Rather, she waits until he grows a little and gets strong.

Tosfos (Menachos 62b DH v'Chen): In Pesachim, why do we permit an Arel or Tamei to send any Korban (except for Pesach) through a Shali'ach? He loses the Mitzvah of Semichah! We can say that we discuss people without a way to fix this. The Arel's brothers died due to Milah. The Tamei is a Zav or Metzora. It is not in his control to become Tahor (until the Tzara'as or Zivah ceases).

Nimukei Yosef (Yevamos 20a DH u'Shlishi): If a woman's first two sons died due to Milah, we do not circumcise the third on the eighth day, until he strengthens.

Rashba (Yevamos 70a DH ha'Arel): Rashi explains that "Arel" refers to one whose brothers died due to Milah. The Gemara said that an Arel Kohen may eat, just his mouth hurts him. I.e. he is destined to (eat after he) circumcise(s) when he gets strong and his blood is absorbed. R. Tam explains that it refers to one who wants to fulfill Mitzvos, but he fears the pain of Milah, and therefore he refrains. Rashi's Perush is primary.

Chasam Sofer (YD 245): If there is an Arel who never circumcises, the Gemara's question returns (why does he permit others to eat?)! Rather, we always wait for Hash-m's salvation, that we will see signs of health. Even if his siblings died due to Milah in maturity, he can wait until he is older. Perhaps it will be good for him to circumcise in old age.

Note: If all his older brothers died due to Milah at various ages from eight days until 119 years, I cannot fathom when we would expect to circumcise him. If we never circumcise him, the question returns!

Chasam Sofer: In any case most do not die without a Bris. They become healthy before dying, and are circumcised. If not, we would not say that he may eat (just his mouth hurts him), since it is a Safek if he will ever eat. Even according to the Poskim who allow one to be Moser Nefesh for a light Mitzvah and praise this, there is a clear proof from here (that he is obligated to circumcise later). If you would say that there is no limit for an Arel to circumcise, why do we say that he eats? Even if he wants to be Moser Nefesh, it seems that he will die, due to the Chazakah, so we would not say that he eats. It is not difficult to say that the verse discusses Kohanim whose brothers died due to Milah in adulthood. At the time of the Churban there were many who did not circumcise - "u'Vris Kodesh Ya'avru"! (Menachos 53b) Perhaps they repented and circumcised in adulthood, and many died, and the verse forbids their brothers to serve.

Poskim

Shulchan Aruch (YD 263:2): If a woman's first two children died due to Bris Milah, she is established that her children die due to Milah.

Beis Yosef (DH Ishah): It seems that the Rif, Rambam and Rosh say not to circumcise even when the third is not sick at all. Rather, some families have weak blood. If the baby is sick, we do not circumcise him even if he is the first! The Rambam and Tur say that the first two died due to Milah, i.e. and not due to another sickness.

Noda bi'Yehudah (2 YD 165): A case occurred in which a three-year was still not circumcised, because the first two babies died through Milah on day eight. I would have said that we never circumcise him, but the Rambam, Tur and Nimukei Yosef say that we circumcise him after he grows and strengthens. The Gemara said Stam (that we do not circumcise the third).

R. Akiva Eiger (2, citing Maharit 2 YD 31): If the first two babies died through Milah on day eight, and the third baby lived, perhaps the Chazakah is Batel from now and onwards. This is unlike Vestos and a Mu'ad ox. (There, the Chazakah is Batel only after it failed the same number of times needed to establish it.) Those depend on fixing a time.

Sho'el u'Meshiv (1:1:238): If two twins died due to Milah, do we refrain from circumcising the third? Or perhaps, this is only when two babies from different pregnancies died? Seemingly, it should not make a difference. However, we can say the nature of twins is to be weak, for one drop of semen was divided into two. Perhaps this is why they died! In any case it is a Safek, and we are lenient about any number of Sefekos regarding mortal danger (OC 329:3).

Shevet ha'Levi (2:128): Yashresh Yakov (Yevamos 64b DH d'Tanya) proved that Tosfos in Gitin 28b, Bechoros 61a, Metzi'os 75a and Menachos 62b holds that if his brothers died due to Milah, he never circumcises. This is unlike the Rashba and Chasam Sofer. Tosfos asked why an Arel and Tamei may send Korbanos and omit the Mitzvah of Semichah. Tosfos answered that we discuss people without a way to fix this, i.e. his brothers died due to Milah, so he can never circumcise, or a Zav or Metzora, for whom becoming Tahor is out of his control (until the Tzara'as or Zivah ceases). Tosfos' words connote that that he can never circumcise. One could dispel the proof, and say that we do not know when he will become stronger, so he is like a Zav or Metzora, who is not in control of becoming Tahor. Therefore, he may offer without Semichah. However, regarding Temei'im Tosfos said that the cure does not depend on him, and regarding Milah Tosfos said that it is impossible. This connotes like Yashresh Yakov, that we do not circumcise him at all.

Shevet ha'Levi: In practice, it seems that we are stringent like Rashi, that an Arel whose brothers died may not eat Terumah or Kodshim or do Avodah. However, we are stringent not to circumcise him later, like R. Tam.

Shulchan Aruch (ibid.): We do not distinguish whether they had the same or different fathers. She does not circumcise the third until he grows and gets strong.

Beis Yosef (DH u'Mah): The Gemara did not mention whether the babies were from the same father. This connotes that it does not depend on this.

Shulchan Aruch (ibid.): The same applies if a man's first two children died due to Bris Milah. He does not circumcise the third until he grows and gets strong. This is whether they had the same or different mothers.

Source: Beis Yosef (DH v'Chasav), citing R. Mano'ach.

Rema: Some disagree and say that it does not depend on the father. It depends only on the mother. We are lenient about Safek Nefashos.