Hullin, Chapter One, Mishnah Three
Slaughtering must be performed on the animals neck below the larynx, preferably below the first hard ring of the trachea and up to the place where the bronchial tubes begin to branch. Cutting outside of this area is called hagramah and renders the animal invalid. Our mishnah deals with a case where a person cuts right below the first hard ring of the trachea.
1) If one slaughtered [by cutting] at the [top] ring [of the trachea] and left a hair’s breadth of its entire circumference [towards the head], the slaughtering is valid.
2) Rabbi Yose son of Rabbi Judah says: if there was only left [towards the head] a hair’s breadth of the greater part of its circumference, [the slaughtering is valid].
Section one: As stated in the introduction, the cut should be made below the first hard ring of the animals trachea. If he makes the cut at this top ring, the slaughtering is valid as long as a hairs breadth of the trachea on the side of the head remains. According to the first opinion, the hairs breadth of the trachea must remain on the entire circumference of the highest ring.
Section two: Rabbi Yose son of Rabbi Judah says that it is sufficient for the hairs breadth to remain on the greater part of the circumference, and not necessarily the entire circumference.