Yevamot 50 - Divorce, and then another divorce
If a deceased brother left two wives, and the remaining brother gives each one a Get (divorce document), then the second Get has no validity. Really, the Get has no place here, it should be either a yibum (marriage) or a chalitzah, which is akin to divorce. However, if he does give the first wife a Get, and the Get has no effect, then certainly the Get given to the second wife has no effect either, and he can marry the second wife's relatives (mother, sister, etc.)
If, on the other hand, a Get does have the same effect as chalitzah, then all bonds between the remaining brother and the second wife are already dissolved, and giving her a Get now means nothing, for he is giving a Get to a stranger. This is the opinion of Rabban Gamliel, which can be expressed as "There is no Get after a Get."
The Sages, however, say that "There is Get after a Get," meaning that Get weakens the bond but does not dissolve it completely. Therefore, the first wife still needs a chalitzah, even after she got a Get. Also, the second wife is still connected to the brother, and if he gives her a Get, it is valid to the extent that he cannot marry her relatives, and she cannot marry his.
Why is this partial situation? - This decree is to avoid confusion. If the Get had no validity, people could reason thus: "Get effects separation and chalitzah effect separation. Just as we see that the Get here has no effect, so too chalitzah has not effect other than allowing the woman to marry anyone outside the family," and the people may then do a yibum, which is prohibited after chalitzah. This follows the rule that there is only one chance to re-build the brother's house, but not two chances.
Art: Four Fisherwives by Winslow Homer