Beyond the significance of the firstborn son in the laws of pidyon haben, there is another area of halachah in which being a firstborn carries legal weight: the laws of inheritance. The firstborn takes a double portion relative to his brothers when they inherit from their father, and our Mishnah comes to define precisely when this applies.
The parameters of the firstborn's portion in inheritance:
In the father's property and not in the mother's property - the firstborn takes a double portion only in inheriting his father's property and money. If the mother died after the father and the sons inherit from her, the firstborn divides with them equally and receives a portion identical to that of the other sons.
He does not take a double portion in appreciation - regarding the part of the inheritance that improved and rose in value after the father's death, such as new produce that grew after he died, that portion is divided equally among all the sons.
He does not take a double portion in property that is 'ra'ui' (due to come), but only in property that is 'muchzak' (already held) - with money that was destined to reach the father, the firstborn does not take a double portion the way he does with what already came into his possession. For example, if the father lent money and someone owed him, that person owes the money to the heirs automatically; but since at the time of death the money was not held in the father's possession, the firstborn does not take a double portion in it, and that part of the inheritance is divided equally.
Additional entitlements governed by the same rule:
A woman's kesubah: the kesubah is a document written at the time of the marriage, entitling the wife to a certain sum of money if her husband dies or divorces her. Either way, she collects from her husband's estate only the money that was already there at the time of his death, and not from what improved and rose in value after he died, or from money that arrived only later.
Support for the daughters: the daughters of a man who died are fed and supported from the inheritance money he left behind. Here too, this applies only to the money that was with him at the time of his death, and not to money that came in afterward or that rose in value afterward because of something new that grew.
The yavam: the mitzvah of yibum applies when a man dies without children, and a mitzvah falls upon his brother to marry his widow. When the brother fulfills the mitzvah of yibum and marries her, the law is that he inherits all of the deceased's property, and the other brothers - who ordinarily would also have inherited - do not inherit. However, this applies only to the main body of the inheritance, whereas whatever comes afterward, as in the other cases discussed in the Mishnah, is divided among all the brothers.
In all these cases the Mishnah establishes a single rule: "V'chulan ein notlin ba'shevach u'v'ra'ui k'muchzak" - none of the inheritance laws and entitlements discussed in the Mishnah applies to what grew and appreciated afterward from the estate of the deceased, nor to what was due to come to him but had not yet reached his possession, such as a loan that another person owed him and repaid only after his death.