Hullin, Chapter Two, Mishnah Three
1) If he chopped off the head with one stroke, the slaughtering is invalid.
2) He was slaughtering and he cut through the neck with one stroke, if the knife was as long as the neck, the slaughtering is valid.
a) He was slaughtering and he cut through two necks with one stroke, if the knife was as long as one neck the slaughtering is valid.
b) When is this so? When the slaughterer moved the knife forward and not backward, or backward and not forward; but if he moved the knife to and fro, however small it was, even if it was a scalpel, the slaughtering is valid.
3) If a knife fell down and slaughtered [an animal], even though it slaughtered it in the proper way, the slaughtering is invalid, for it is said, And you shall slaughter and eat, that which you slaughter, you may eat.
4) If [while slaughtering] the knife fell and he picked it up, if his clothes fell and he picked them up, if he sharpened the knife, or if he got tired and his friend came and [continued] slaughtering, if he delayed the time that it takes to slaughter, it is invalid.
a) Rabbi Shimon says: if he delayed the time it takes to examine the slaughtering.
Section one: Slaughtering must be done with a to and fro motion with the knife and not by chopping. Chopping the head off, which is called derasah (pressing), invalidates the animal.
Section two: In this case, while slaughtering, which means bringing the knife back and forth, he pressed the knife into the neck, and cut the organs. If the part of the knife that has already passed through the neck is as long as the neck, then the slaughtering is valid. The reason is that we can assume that he has already slaughtered properly with the part of the neck that was already passed through the neck.
So too, if he slaughters two animals at the same time, and he presses down on one of them. If the length of one knife has already passed through both necks, the slaughtering is valid.
These limitations are necessary if he had passed the knife only in one direction. But if he had already passed the knife in both directions, it doesnt matter how large the knife is, since we can assume that he slaughtered it before he chopped it.
Section three: Slaughtering must be performed by a person slicing the neck, not by a knife that somehow falls down and slices a neck without being held by a person. While this doesnt seem to be a real possibility, perhaps what it means to say is that there must be human involvement in the slaughtering process. It cant just happen on its own, even if it happens in a valid fashion.
Section four: The mishnah lists various reasons that a person might stop in the middle of slaughtering an animal. If he stops for any of these reasons, and doesnt resume slaughtering for the amount of time it would take to slaughter an animal, then the slaughtering is invalid. In other words, the second half of the act of shechitah does not join with the first act and therefore the animal was not fully slaughtered in the correct manner. Stopping in the middle of slaughtering is called shehiyah, delaying, and it is one of the things that renders slaughtering invalid.
Rabbi Shimon says that if he delays the amount of time necessary for someone to check if the slaughtering was done properly, then it is invalid. This is a longer time than it would take to actually slaughter.