Kinnim, Chapter 2, Mishnah 2. The previous mishnah established the rule that an undesignated bird which flies out of one set and lands in another set causes one bird in each of the two sets to become pasul. This mishnah takes that abstract rule and puts it to work with a concrete case, tracking exactly how many birds remain fit as the bird travels back and forth.
The Case
To make the ruling tangible, the mishnah asks "Keitzad", in what sort of situation? It answers with "Shetei nashim", two women who came with their obligations, each of them holding, in the words of the mishnah, "lazo shetei kinnim", this one with a pair and a pair, "velazo shetei kinnim", and the other one likewise with two pairs. So every woman stands with four live birds awaiting sacrifice, and within neither group of four has it yet been decided which bird will serve as the chatat and which as the olah.
The First Flight
The mishnah continues: "Parach mizo lazo", one of the birds abandons the first woman's group and settles among the second woman's birds, and therefore "posel echad bahalichato", its journey costs one bird on each side. Count the birds: the second woman is now holding five, while the first is holding only three. Among those five, a single bird can no longer be offered at all, so four of them remain available. Among the three, one is likewise ruled out, so only two of them remain available.
The Return Flight
"Chazar, posel echad bachazirato." Now a bird flies back from the second set to the first, so that each set once again holds four birds. This second flight disqualifies one more bird in each set, two additional losses in all. Before the return flight, one bird had been disqualified in the second set and one in the first. Now a second bird is disqualified in each. The result: of the four birds in the second set, two are pasul and two may be offered, and of the four birds in the first set, two are pasul and two may be offered. Each woman is therefore left with exactly two usable birds, which she brings as one chatat and one olah. Half of what each woman brought has been lost.
Why Nothing More Is Lost
"Parach vechazar, parach vechazar, lo hifsid kelum." From this point onward, a bird may fly across and return, and fly across and return again, however many times it happens, and no further loss results. The reason: "she'afilu hen me'oravot, ein pachot mishtayim." Even if all the pairs of both women were poured together into a single mixture, at least two pairs would still be fit to be offered. This follows from the principle learned earlier, that when equal sets of pairs become intermingled, half of them remain fit. Here there are four pairs in all, so two pairs may still be brought: any four birds from the combined group are taken, two offered as chatat and two as olah. Since that minimum is guaranteed no matter how thoroughly the birds mix, additional flights back and forth can cost the women nothing more.