Gemara

Question (Ilfa): Can hands become Tahor half-way?

He does not ask about washing one hand at a time, for a Mishnah allows this! (One may pour water on one hand and immerse the other.) Rather, he asks about one who washed half of his hand, and then the other half.

Question: R. Yanai taught that hands do not become Tahor half-way!

Answer #1: Ilfa asks about when there is still liquid Tofe'ach (wet) on the first half when he washes the second half. R. Yanai discusses when the first half dried before washing the second.

Objection (Mishnah): A liquid Tofe'ach is not a connection for Tum'ah or Taharah.

Answer #2: Ilfa asks about when the first half still has liquid Tofe'ach Lehatfi'ach (wet enough to wet something else).

Question: A Beraisa explicitly teaches that Tofe'ach Lehatfi'ach connects!

Answer: Perhaps the Beraisa discusses Mikva'os, and it is like R. Yehudah.

Mishnah (Yadayim 1:1): A Revi'is may be used to wash the hands of one person, or even two. A half-Log (two Revi'iyos) may be used to wash the hands of three or four. A Log may be used to wash the hands of five, 10 or 100;

R. Yosi says, a Revi'is must remain for the last person.

One may add to Mayim Sheniyim, but not to Mayim Rishonim.

Rishonim

Rosh (Yadayim 1:1): A Log is a large amount. The first Tana allows as many as the water suffices for; 100 is an exaggeration. R. Yosi argues about a Log, lest there not remain enough for the last person. Therefore, he requires for the last person the Shi'ur (a Revi'is) for Netilah (washing).

Rosh (ibid.): The first water becomes Tamei, and the second water is Metaher it. If the Rishonim did not go until the Perek, pouring more only increases Tum'ah. He must dry the hands and start again. One can add to Sheniyim. R. Meir argues in a Tosefta. He permits adding to Rishonim, but not to Sheniyim. Sheniyim are Metaher only if poured at once. The Halachah follows our Stam Mishnah. Perhaps one may add to Rishonim and, according to R. Meir, to Sheniyim, if he washes the entire hand until the Perek. Washing cannot be half-way, for people do not wash this way. The Gemara did not try to settle Ilfa's question from our Mishnah, because R. Meir and R. Yosi argue in the Tosefta.

Rambam (Hilchos Mikva'os 11:7): If one washed part of his hand and later washed the rest, it is Tamei like beforehand. If the first half was still Tofe'ach Lehatfi'ach when he washed the rest, it is Tahor.

Rebuttal (Ra'avad): He erred. The Halachah follows R. Yosi against R. Meir. One may not add to the Rishonim; there must be a Revi'is. Tofe'ach Lehatfi'ach does not help, for there was no Shi'ur Netilah. Perhaps the Rambam rules like R. Meir in the Tosefta (Yadayim 1:1), for it was taught Stam in the Mishnah (1:1) that we do not need a Revi'is for the last person.

Kesef Mishneh: The Ra'avad explains that Tofe'ach Lehatfi'ach helps only when there was a Revi'is at the beginning. Or, he understands that the Rambam does not require a Revi'is for the latter part of the hand, since he did not mention this. The Rambam (Halachah 8) requires a Revi'is for each person. He holds like our Stam Mishnah, which permits adding to Sheniyim, but not to Rishonim. However, if the hand is Tofe'ach Lehatfi'ach, one may add even to Rishonim, like it says in Gitin. This is like one washing!

Beis Yosef (OC 162 DH u'Kevar): The Rambam does not require Sheniyim for Chulin. He holds that the Mishnah discusses Terumah; in Hilchos Berachos he did not forbid adding to Mayim Rishonim. The Rambam agrees with the Rosh, that one may always add if he washes the entire hand at once.

Rambam (ibid.): This refers to Mayim Rishonim. For Sheniyim, one may wash part of the hand, and then wash another part.

Rebuttal (Ra'avad): He erred about this also. The Mishnah says 'one may add to Sheniyim, but not to Rishonim.' The Tosefta explains that if he washed Rishonim, rubbed his hands, and the Sheniyim did not reach the Perek, he may add water. The Rambam holds that this is even if the hands dried in between. This is wrong. Even R. Meir, who does not require a Revi'is for the last person, does not allow this. Hands cannot be washed half-way at all.

Rosh (ibid.): The Rambam explains that only Sheniyim can be from less than a Revi'is. No one permits this for Rishonim. The Halachah is unlike R. Yosi.

Poskim

Shulchan Aruch (OC 162:3): If one washed part of his hand and later washed the rest, it is Tamei like beforehand, for washing cannot be half-way. If the first half was still Tofe'ach Lehatfi'ach when he washed the rest, it is Tahor.

Beis Yosef (DH u'Mah she'Chosav she'Im): We did not settle whether or not Tofe'ach Lehatfi'ach helps for Netilas Yadayim. The Rambam rules that it does, for we are lenient about Safek Netilas Yadayim. Why didn't the Rosh say so? Perhaps he is Metaher hands regarding a Safek whether or not the water was used for work, but not for a Safek Din (an unsettled question). However, normally we are lenient even about a Safek Din mid'Rabanan!

Taz (5): R. Tam (Chagigah 21b DH k'Ovyah) infers (regarding Mikva'os) that Tofe'ach does not help, but Tofe'ach Lehatfi'ach helps! She'ivah disqualifies a Mikveh mid'Rabanan, yet the Rosh, Tur and Rema (YD 201:53) say that Tofe'ach Lehatfi'ach does not suffice for this. We should likewise be stringent about Netilas Yadayim. Surely, a flow that does not cease is like one pouring.

Mishbetzos Zahav (5): A 'Teiku' cannot be resolved, so we are lenient about a Sefek-Sefeka (two doubts) or a Safek mid'Rabanan. We are not lenient about these when the Gemara does not say Teiku. Then, we lack knowledge. Perhaps the question was settled elsewhere, or a Chacham will settle it!

Magen Avraham (5): It is b'Di'eved to rely on Tofe'ach Lehatfi'ach. This is why Ilfa could not settle his question from the Mishnah. It forbids l'Chatchilah with Tofe'ach. Ilfa asked about Tofe'ach Lehatfi'ach, b'Di'eved.

Mishnah Berurah (30): If one washes from a pump and must open and close it for each pouring (so the water will come from human impetus), this is not an interruption (it is like one pouring). Nevertheless, if one has another Kli, he should not wash from such a pump.

Kaf ha'Chayim (18) and Chazon Ish (24:8): Just like Tevilah cannot be half-way, also Netilah. This is even if he pours a Revi'is on half the hand, and another Revi'is on the rest.

Kaf ha'Chayim (19): The Rosh requires drying the hands after pouring on part; the Poskim do not bring this. We always wash three times, so it suffices to wash the entire hand twice afterwards. One who is stringent to dry it first will be blessed. If he rubbed his hands together, he is obligated to dry them.

Shulchan Aruch (ibid.): This refers to Mayim Rishonim. For Sheniyim, one may wash part of the hand, and then wash another part.

Mishnah Berurah (31): This is even if the hands dried in between.

Kaf ha'Chayim (21): One with Yir'as Shamayim should wash the entire hand every time to fulfill all opinions.

Beis Yosef (DH v'Efshar): The Tur cites an opinion that Rishonim is before the meal, and Sheniyim is after the meal. Perhaps he refers to R. Shimshon, who cited the Yerushalmi, which explains like this. I say that R. Shimshon merely proved from the Yerushalmi that one who did not wash until the Perek must dry his hands and wash again.

Mishnah Berurah (25): Some say that the Perek is the where the fingers meet the palm. One should be stringent like the opinion that it is the wrist (161:4).