This mishnah, in Maseches Eduyos chapter 3, mishnah 5, deals with the sling - the implement used by a shepherd, and the very implement with which David slung the stone at Goliath. The question under discussion is one of receiving tumah: is the sling, and in particular its middle section, considered a vessel that can receive tumah?
The structure of the sling:
The pouch (beis kibbul) - the central and main part, a sort of pocket in which the stone is placed. In truth it is nothing more than a flat piece of cloth or leather.
The finger loop (beis etzba) - a cord extending from one side of the pouch, with a loop at its end that goes around the finger of the one slinging.
The release cord (beis hapki'a) - a cord extending from the other side, with a small knot at its end that is held and then let go at the moment of the throw.
How it works: the stone rests in the pouch, both cords are held in the hand of the one slinging, he whirls the sling and releases the beis hapki'a - and the stone flies off. This is a weapon in every sense.
The rules of receiving tumah for cloth and leather:
A plain piece of woven cloth - a cut piece that has no particular purpose does not receive tumah unless it measures three fingerbreadths by three fingerbreadths. This measure was set because anything smaller is not even fit to serve as a patch to repair a hole in a garment.
Woven cloth that has been put to use - anything made from it that has a purpose and a usefulness receives tumah at any size, even smaller than this measure.
Plain leather - leather that is not a vessel has a measure of five handbreadths by five handbreadths for receiving tumah.
Therefore, a pouch made of woven material - even though it is nothing more than a flat piece, and even if it does not measure three fingerbreadths by three fingerbreadths - is a vessel that holds the stone, and together with the cords on its sides the sling constitutes a complete, functional implement. It therefore receives tumah according to all opinions, and not the pouch alone but the cords on the sides as well, since they are subordinate to it.
A pouch made of leather:
What is the law when the pouch is not woven but rather a small strip of leather? When the sling is not in use, all we have before us is a plain strip of leather with two cords extending from its sides, and its measure is far less than five handbreadths by five handbreadths. Does it receive tumah?
The wording of the mishnah: "HaKela shebeis kibbul shelah arug - temei'ah; veshel or - Rabbi Dosa ben Harkinas metaher, vaChachamim metam'in" - a sling whose pouch is woven is tamei; and one of leather - Rabbi Dosa ben Harkinas rules it tahor, and the Sages rule it tamei.
Rabbi Dosa ben Harkinas rules it tahor - all we have here is a small strip of leather that does not have the required measure, and it is not considered a vessel at all.
And the Sages rule it tamei - in the context of the sling, even this flat strip is a full-fledged pocket, which holds the stone and actually serves for slinging by means of its two sides. It is a vessel, and it receives tumah.
And the halachah follows the Sages.
If one of the cords snaps:
The Mishnah continues and discusses a sling that has broken: "Nifsak beis etzba shelah - tehorah; beis hapkia - temei'ah" - if its finger loop snapped, it is tahor; if the knot end snapped, it is tamei. Is the utensil considered broken, so that it no longer receives tumah?
If the finger loop snapped - the loop that goes around the finger is essential to the working of the sling, since without it the utensil cannot be held and the sling cannot be swung effectively. Once it has snapped, the sling is no longer fit for its work, and therefore it is tahor.
If the knot end snapped - the knot at the end of the second cord serves only for convenience in gripping and releasing, and it can simply be tied again. Even if the cord has become slightly shorter, that is of no consequence, since it can be lengthened. The utensil remains fit for its use, and therefore it still receives tumah.
In summary: we have learned the structure of the sling and its law with regard to tumah. A woven pouch is tamei according to all opinions; regarding a pouch made of leather, Rabbi Dosa ben Harkinas and the Sages disagree, and the halachah follows the Sages that it too is a utensil and receives tumah. And where one of the cords snapped: if the finger loop snapped it is tahor, since the utensil is no longer fit for its work; if the knot end snapped it is tamei, since the utensil is still fit for its use.