Keilim, Chapter 25, Mishnah 5. This chapter is built around a single principle: in the laws of tumah, a vessel is not always treated as one indivisible object. Its inner space and its outer surfaces can carry separate statuses, so that tumah which reaches one of them does not automatically spread to the other. Our mishnah applies that principle to a vessel that was manufactured with two compartments of unequal size, one with a capacity of a rova and one with a capacity of half a rova. The previous mishnah dealt with what happens to one compartment when the other one contracts tumah. Here the discussion moves to the walls of such a vessel: which outer surfaces belong to which compartment, and how far the tumah of an outer surface travels.
When the inside of one compartment becomes tamei
The mishnah opens: "Nitma harova, harova va'achorav temei'in." If tamei liquid entered the inner space of the rova compartment and rendered it tamei, then that compartment together with its outer surfaces is tamei. This is the ordinary rule: when tumah reaches the inside of a vessel, the whole vessel is affected, its outside included.
The mishnah continues: "Chatzi harova va'achorav tehorin." The inner space of the half-rova compartment and the outer surfaces belonging to it remain tahor. Although the two compartments are physically joined in one utensil, for this purpose each is regarded as a container in its own right, and the tumah of one does not carry over to its neighbor.
Which walls belong to which compartment
Since each compartment is credited with outer surfaces of its own, we must know where the boundary between them falls. A compartment like this is bounded on three sides by walls that serve it alone, and on the fourth side by the partition it shares with its neighbor. Everything external to a compartment's own three walls counts as its achorayim, and the identical reckoning applies to its counterpart. The shared partition in the middle is apportioned by thickness: half of it is credited to the one compartment and half to the other.
The reverse case
The mishnah then states the mirror image: "Nitma chatzi harova, chatzi harova va'achorav temei'in, harova va'achorav tehorin." If it was the inner space of the half-rova that came into contact with tamei liquid, then the half-rova and the outer surfaces belonging to it are tamei, while the rova and its outer surfaces stay tahor. The rule works identically in either direction.
When tumah touches only an outer surface
The mishnah now shifts to a different case: the tamei liquid made contact not with a cavity but with the exterior wall of the rova. In such a situation the inner space of the rova is unaffected, for a keli that becomes tamei on its back does not pass that tumah inward. And once the cavity of the very compartment that was touched stays tahor, it is self-evident that the cavity of its neighbor, the half-rova, stays tahor as well.
All that is left to determine is how far the tumah spreads across the back of the utensil, and here the Tannaim part ways. The mishnah records: "Nitme'u achorei harova" - liquid rendered the exterior of the rova tamei - and the ruling follows, "achorei chatzi harova tehorin", the exterior belonging to the half-rova is unaffected, "divrei Rebbi Meir". Rebbi Meir holds to the principle we have been following throughout: although one utensil houses both compartments, halachah looks upon them as two independent receptacles, so tumah on the one does not migrate to the other.
The view of the Chachamim
The Chachamim rule otherwise: "Vachachamim omrim", and their words are "ein cholkin et hagabayim", the backs are not subject to any division. They readily accept that the cavities divide, for there the eye sees two distinct receptacles, each holding a status of its own. The exterior, though, is one unbroken shell wrapped around the entire utensil, with nothing to apportion. The result is that the instant liquid renders the back of one compartment tamei, the back of the second is tamei together with it. The mishnah adds: "ukeshehu matbil", when the owner comes to immerse such a keli, "matbil et kulo".
The Chachamim add a practical consequence. When the owner comes to purify this utensil in a mikveh, he must immerse the entire utensil. Even though only one part of it contracted tumah, and even though the two compartments are treated as distinct containers, a vessel cannot be purified unless the whole of it enters the water. So in a case where we rule that the whole utensil is tamei and the owner wishes to restore all of it to taharah, that is achieved only by immersing the utensil in its entirety.