Beitzah, chapter three, mishnah three. The mishnah deals with someone who has a dangerously ill animal on Yom Tov - an animal that is at risk of dying. As long as it is alive, the owner wants to slaughter it so that he will be able to use its meat, since if it dies on its own it becomes a neveilah and will not be fit to eat. The problem is that he has already eaten his meal, and seemingly he does not need the animal for eating purposes - and the permission to slaughter on Yom Tov was granted only for the sake of eating on Yom Tov, and not for the sake of what will be afterward.
The words of the mishnah:
"Beheimah mesukenes lo yishchot, ela im ken yesh shehus bayom le'echol mimenah kezayis tzli" - a dangerously ill animal may not be slaughtered unless enough time remains in the day to eat at least a kezayis of roasted meat from it. Roasting is the easiest way to prepare the meat in a short time, since there is not enough time to salt it; when roasting, the meat cooks and the blood is drawn out of it at one and the same time. If there is enough time to obtain at least this amount of meat, it is permitted to slaughter the animal.
"Rabbi Akiva omer: afilu kezayis chai mibeis tevichasah" - in his opinion there is no need for enough time to eat a kezayis of roasted meat; it is enough that there be time to eat a kezayis of raw meat from the place of slaughter, which is the most available and convenient area to take from.
As for the blood, according to Rabbi Akiva there is no concern, because the blood that is forbidden is blood that has moved from its place, even if it moved within the animal itself. When the meat is cooked, the blood moves from area to area and becomes forbidden; but one who eats the meat immediately after the slaughter does not encounter this problem. Therefore, this minimal amount of time is enough to permit slaughtering the animal on Yom Tov.
"Shachat basadeh":
The mishnah continues with someone who slaughtered an animal in the field and wants to bring it to his home. The commentators disagree about how to understand the case:
Some understand that it is again speaking of an animal that is at risk of dying.
And some understand that it is speaking of an ordinary animal that was slaughtered in the field.
"Lo yevi'enah bemot uvemotah" - he may not bring it on poles, whether large poles or small ones, because this way of carrying looks like the way loads are transported on weekdays, and it creates a great commotion around the matter, so that he is found to be diminishing the honor of Yom Tov through this sort of activity.
Therefore the mishnah says that he should bring it by hand: he should bring one limb or a few limbs at a time, and conduct himself in a minimal fashion, and not turn it into a large procession of two or more people carrying it on poles and the like.
In summary: in this mishnah we learned that a dangerously ill animal is not slaughtered on Yom Tov unless enough time remains in the day to eat a kezayis of roasted meat from it, and according to Rabbi Akiva it is enough that there be time to eat a kezayis of raw meat from the place of slaughter - and there is no concern about blood, since the prohibition of blood applies to blood that moved from its place during the cooking. We also learned that one who slaughters in the field may not bring the animal on poles, out of respect for the honor of Yom Tov, but should carry it by hand, limb by limb.