Maseches Pesachim, Chapter 8, Mishnah 3. The Mishnah deals with a father who says to his sons before they go up to Jerusalem to slaughter the Pesach offering: "Hareini shochet es haPesach al mi she'ya'aleh mikem rishon liYerushalayim" - I am hereby slaughtering the Pesach on behalf of whichever one of you reaches Jerusalem first, meaning that the offering will be slaughtered in the name of that son who arrives in Jerusalem first.
What the father intended by his words:
The Gemara explains that the father did not mean to register only his quickest son for his Pesach offering, but rather to spur his sons on and encourage them to hurry and go up to Jerusalem. In truth he included all of them in the Pesach offering, and he used this expression only so that they would all hurry. Therefore, once the first son arrived and brought in "rosho v'rubo" - his head and the greater part of his body - he acquired his portion in the Pesach offering, and through him his brothers acquire along with him, since they too are included in the offering.
The minimum amount for those registered for the Pesach:
The Mishnah establishes a rule: one may always register people for the Pesach offering and add more participants, provided that there is a k'zayis of meat in the offering for each and every one of them.
Until when may one register, and until when may one withdraw:
From here the Mishnah discusses two time limits in the process of bringing the Pesach offering:
Up to what stage may a person register for a particular Pesach offering.
Up to what stage may a person withdraw from a particular Pesach offering in order to join a different one, since a person cannot be registered for two offerings at once.
According to the Tanna Kamma, both are possible until the moment of the slaughter. Once the offering has been slaughtered, one can no longer join it, and one who was registered for it can no longer retract, and he remains registered for that offering.
Rabbi Shimon disagrees: regarding withdrawal, a person may withdraw even after the slaughter, as long as the blood has not yet been sprinkled on the altar. Once the blood has been sprinkled, the mitzvah has already been fulfilled through his registration for that offering, and there is no longer any possibility of registering for a different Pesach offering. Regarding registration, however, Rabbi Shimon agrees with the Tanna Kamma that one may register only until the slaughter, and he disagrees only regarding withdrawal.
In summary: in this Mishnah we learned the law of one who says to his sons, "I am hereby slaughtering the Pesach on behalf of whichever one of you reaches Jerusalem first" - that his intention is only to spur them on, that all of them are registered for the offering, and that the first one to bring in his head and the greater part of his body acquires his portion and confers it upon his brothers along with him. We also learned the rule that one may add participants as long as there is a k'zayis for each one, and the time limits for registering and for withdrawing: according to the Tanna Kamma, until the slaughter in both cases, and according to Rabbi Shimon, withdrawal is possible until the sprinkling of the blood.