Gemara

18a (Mishnah): If one slaughters in the ring (around the Kaneh), if the entire cut was within the ring, it is Kosher;

R. Yosi b'Rebbi Yehudah is Machshir if the majority of the ring was cut before the knife left the ring.

(Rav and Shmuel): The Halachah follows R. Yosi b'Rebbi Yehudah.

19a - Version #1 (Rav Huna citing Rav Asi): Chachamim and R. Yosi b'Rebbi Yehudah argue when the first two thirds were slaughtered, and the last third was Hagramah (cut in a place invalid for Shechitah, i.e. above the top ring);

Chachamim require the entire Shechitah in the ring. R. Yosi requires only the majority.

If the first third was Hagramah, and the last two thirds were slaughtered, all disqualify, for the Shechitah is completed when the majority is cut, and at that point, there was more Hagramah than Shechitah.

Question (Rav Chisda): Just the opposite! They argue when the first third was Hagramah, and the last two thirds were slaughtered. R. Yosi is Machshir, just like when half the Kaneh was cut before the Shechitah. Chachamim say that Hagramah is worse, for the Kaneh was not cut in the place of Shechitah;

If the first two thirds were slaughtered, and the last third was Hagramah, all are Machshir.

Version #2 (Rav Huna): Chachamim and R. Yosi b'Rebbi Yehudah argue in a case that the first third was Hagramah, and the last two thirds were slaughtered (like Rav Chisda explained above);

If the first two thirds were slaughtered, and the last third was Hagramah, all are Machshir. (end of Version #2)

(Rav Huna citing Rav): If a third was Hagramah, a third was slaughtered, and the last third was Hagramah, this is Kosher;

Shechitah is complete when the majority is cut, and this was in a valid place for Shechitah.

(Rav Yehudah citing Rav): This is Treifah, for the majority must be Shechitah.

(Rav Yehudah citing Rav): If a third was slaughtered, a third was Hagramah, and the last third was slaughtered, this is Kosher;

(Rav Huna): This is Treifah.

Rav Chisda (to Rav Huna): This is like your first teaching, in which you were Machshir because the cut that made the majority occurred in a valid place of Shechitah. Here also, it occurred in an invalid place, so it is Treifah!

(Rav Nachman): If a third was slaughtered, a third was Hagramah, and the last third was slaughtered, this is like R. Elazar bar Minyomi taught;

(R. Elazar bar Minyomi): Shechitah like a comb (zigzag) is Kosher.

Rejection: Perhaps that is only when all the cutting was Shechitah.

(Rav Yehudah): If a third was slaughtered, a third was Hagramah, and the last third was slaughtered, it is Kosher. If a third was Hagramah, a third was slaughtered, and a third was Hagramah, it is Treifah.

20a (R. Yirmeyah): Whatever is Kosher for Shechitah, correspondingly on the Oref is Kosher for Melikah.

Inference: Whatever is invalid for Shechitah is invalid for Melikah.

(Rav Papa): He teaches that Melikah is invalid on the (bottom of the) back of the head;

(Rav Huna): If the first third was Hagramah, and the last two thirds were proper Shechitah, it is invalid.

Rishonim

Rambam (Hilchos Shechitah 3:13): If one slaughtered the majority of one or both Simanim, and finished the Shechitah through Drasah or Hagramah, it is Kosher, for the required amount was slaughtered properly. If one did Hagramah on the first third and slaughtered two thirds, it is Kosher. If one slaughtered a third, did Hagramah on a third, and slaughtered the last third, it is Kosher. If one did Hagramah on a third, slaughtered a third, and did Hagramah on the last third, it is Pasul. If he did Drasah or Chaladah in the first or middle third, it is Pasul.

Rosh (1:24): Most Poskim rule like Rav Yehudah, like the second version of Rav Huna, in which R. Yosi is Machshir when he did Hagramah on the first third and slaughtered two thirds. Also Rav Nachman holds like Rav Yehudah. However, below (30b) we asked about who did Chaladah on a minority of the Simanim. Rashi (30b) explained that after the majority was slaughtered properly, do we say that it is Kosher (even though Chaladah occurred later)? Or, perhaps it is all one Shechitah, and there is Chaladah! It was obvious to Rashi that Chaladah in the first half of Shechitah makes a Neveilah. The question was not resolved, so we are stringent. Also regarding Hagramah, whether he slaughtered a third, did Hagramah on a third, and slaughtered a third, or did Hagramah on a third, slaughtered a third, and did Hagramah on the last third, or slaughtered two thirds and did Hagramah on a third, we are unsure. Perhaps we hold like Rabanan who disqualify all of these. We are stringent. Above, we said that if one paused after slaughtering half the Kaneh, even if we consider half like the majority, the pause does not disqualify. That was a mere rejection; we do not rely on it. Really, half is unlike the majority; cutting the first half of the Kaneh does not make a Tereifah. Bahag says that pausing in the first half (of the Kaneh) or the second half of (either of) the Simanim disqualifies. Since the question was not resolved, we are stringent. All this is from Rashi. This is astounding. All agree that if the first two thirds were slaughtered and a third was Hagramah, it is Kosher! The Rivam distinguishes between Hagramah, and Chaladah and pausing. Regarding the latter two there is a Safek if they occurred after slaughtering half. Regarding Hagramah, we surely follow the majority, since Hagramah is not in the place of Shechitah. It is like cutting the leg after Shechitah. This is reasonable. The Rambam totally permits.

Ran (5b DH v'Shachat): Rav Huna holds that Hagramah is like a place proper for Shechitah. It is like Shechitah done improperly, like when half the Kaneh was cut before Shechitah. Therefore, it depends only on cutting the majority of the Kaneh. If this was done through Shechitah, it is Kosher. If it was completed through Hagramah, it is Pasul. This is why he is Machshir when he did Hagramah on a third, slaughtered a third, and did Hagramah on a third, and disqualifies if he slaughtered a third, did Hagramah on a third, and slaughtered a third. Rav Yehudah holds that Hagramah is not like a place proper for Shechitah. Rather, it is like a place of (making) Tereifus. If the majority was Hagramah, it is Pasul. If the majority was not Hagramah, it is not Asur due to that cut. However, since it is not a place of Shechitah, most of the Kaneh must be slaughtered, whether at the beginning or at the end. This is why he disqualifies when one did Hagramah on a third, slaughtered a third, and did Hagramah on a third, and is Machshir if he slaughtered a third, did Hagramah on a third, and slaughtered a third. The Rambam rules like Rav Yehudah. The Ba'al ha'Ma'or and Ramban say that there is no way to decide, therefore we are stringent like both opinions. We are Machshir only if most was slaughtered at once, whether at the beginning or at the end. Rav Yehudah and Rav Huna agree that this is Kosher, for most was slaughtered, and it died through Shechitah. The Rashba is more stringent. He rules like the first version of Rav Huna in the name of Rav Asi. We are Machshir only if the first two thirds was Shechitah and the last third was Hagramah. If the first third was Hagramah and the last two thirds was Shechitah it is Pasul, for most must be Shechitah at the time it dies, and this is not fulfilled. He brought a proof from 20a, which says that R. Yirmeyah's words include the law of Rav Huna in the name of Rav Asi.

Beis Yosef (YD 24 DH u'Mah she'Chasav u'Ba'inan): Also Tosfos (30b DH Hichlid) is stringent like the Rashba.

Poskim

Shulchan Aruch (YD 24:13): If one did Hagramah on the first third of the Kaneh and slaughtered two thirds, it is Kosher.

Gra (Likut): This is like R. Yosi b'Rebbi Yehudah, like Version #2 (of Rav Huna). Rav and Shmuel rule like it (18a). Rav Yehudah and Rav Huna agree that it is Kosher, for the majority was through Shechitah, and it died amidst Shechitah. Even though 20a connotes like Version #1, that is according to Rabanan.

Shulchan Aruch (ibid.): If one slaughtered a third, did Hagramah on a third, and slaughtered the last third, it is Kosher.

Shach (16): It is Kosher in these two cases because the majority was slaughtered.

Beis Yosef (DH u'Mah she'Chasav Rabeinu): The Tur says that the Rosh leans to rule like the Rambam. This is based on his initial words 'most rule like Rav Yehudah. Also Rav Nachman holds like Rav Yehudah.' The Rosh did not explicitly say how he rules. Since he said that most rule like Rav Yehudah, and he did not disagree, it seems that he leans to agree. Also Ba'al ha'Itur rules like Rav Yehudah. R. Yerucham says that if one slaughtered a third, did Hagramah on a third, and slaughtered a third, most Poskim disqualify. This is wrong. The Rosh testified unlike this! L'Halachah, since the Rambam, Ba'al ha'Itur and most Poskim rule like Rav Yehudah, and it seems that the Rosh agrees, we follow them.

Shulchan Aruch (ibid.): If one did Hagramah on a third, slaughtered a third, and did Hagramah on the last third, it is Pasul.

Shach (17): This is because the majority was Hagramah.

Shulchan Aruch (ibid.): If he did Drasah or Chaladah in the first or middle third, it is Pasul.