Gemara

Beraisa - R. Akiva: "V'Hizah ha'Tahor Al ha'Tamei*" - Haza'ah is Metaher a Tamei, and it is Metamei a Tahor.

Chachamim: This refers only to something that is Mekabel Tum'ah:

(Mishnah Parah 12:3): If one intended to sprinkle on an animal (which is not Mekabel Tum'ah) and sprinkled on a person, if water remains on the Ezov he may sprinkle again (with intent for the person);

If he intended to sprinkle on a person and sprinkled on an animal, if water remains on the Ezov he may not sprinkle again.

Question: What is R. Akiva's reason?

Answer: It could have said V'Hizah ha'Tahor Alav - Al ha'Tamei is extra to teach that the opposite applies to Haza'ah on a Tahor.

Chachamim: No, it teaches about Haza'ah on something that is not Mekabel Tum'ah. A Kal va'Chomer teaches unlike your law!

If a Tamei received Haza'ah he is Tahor, all the more so a Tahor who received Haza'ah is Tahor!

R. Akiva: (Indeed, it seems illogical!) This is why Shlomo said "V'Hi Rechokah Mimeni" (even he could not understand this)!

Continuation of Mishnah (12:3): Water that drips off is Kosher, therefore it is Metamei like Mei Chatas.

Tosefta (Parah 12:6, according to Hagahos and Bi'ur ha'Gra): If one was Tovel the Ezov (in Mei Chatas) to sprinkle on something that needs Haza'ah (e.g. a Yisrael, and did so), water that drips (off the Ezov while sprinkling) is Tamei and Kosher for Haza'ah; if water remains on the Ezov he may sprinkle it.

If he intended for a person but the drops landed on something that does not need Haza'ah (e.g. an animal), water that drips is Tamei and Pasul for Haza'ah; he may not sprinkle water left on the Ezov.

If he was Tovel with intent for an animal, water that drips is Tamei and Kosher for Haza'ah;

Rebbi says, since the Tevilah was Pasul the Haza'ah is Pasul.

Rishonim

Rambam (Hilchos Parah Adumah 15:1): If someone touched or carried Mei Chatas not for the need of Haza'ah, he becomes Tamei mid'Oraisa.

Rambam (ibid. 10:8): Haza'ah is Metaher only if he intended to sprinkle on the person or Keli. If one was Tovel the Ezov intending to sprinkle on a person and sprinkled on an animal, he may sprinkle water remaining on the Ezov to Metaher people or Kelim, for the Tevilah was Kosher;

If he was Tovel the Ezov to sprinkle on an animal and sprinkled on a person, the Tevilah was Pasul. To sprinkle on people or Kelim he must first Tovel the Ezov with this intent.

Source (Perush ha'Mishnayos 12:3):" V'Hizah ha'Tahor Al ha'Tamei" teaches that intent for something that is Mekabel Tum'ah is required.

Objection (Ra'avad): The Tosefta teaches the contrary! Rather, if he was Tovel and sprinkled with intent for a person and did so, water that drips is Tamei and Kosher for Haza'ah; he may sprinkle water left on the Ezov. If he intended for a person but sprinkled on an animal, water that drips is Tahor and Pasul for Haza'ah; he may not sprinkle water left on the Ezov. There are variant texts of the Mishnah; we can explain the Mishnah according to the Tosefta.

Kesef Mishneh: The Rambam learned from the Tosefta. He had a different text than the Ra'avad, or just explained it differently.

Rambam (ibid. 9): If he was Tovel to sprinkle on an animal, water that drips off the Ezov is Kosher. Therefore, if it dripped into a Keli he may Tovel the Ezov in that water and do a proper sprinkling.

Tosfos Yom Tov (Parah 12:3 DH ha'Mayim): Even though he may not sprinkle drops remaining on the Ezov, when they drip into a Keli and he is Tovel the Ezov in them it is Kosher.

Rambam (ibid. 15:1): If he was Tovel to sprinkle on an animal, water that drips is Kosher. Therefore, it has Tum'as Mei Chatas, because it was not used for the Mitzvah.

Ra'avad: This is wrong, like I explained above (8:8, even if he was Tovel intending for a person, if he sprinkled on an animal even water left on the Ezov is Pasul. This implies that all the more so, Tevilah with intent for an animal disqualifies the water.)

Rashi (DH Lo Yeshaneh): Haza'ah on an animal is not Haza'ah; it disqualifies the water as if it was used for Melachah. (Haza'ah on a person does not Posel, even if he was Tovel intending to sprinkle on an animal - DH ked'Tanan.)

Question: A Mishnah (Parah 4:4) teaches that the Pesul of Melachah applies only until the ashes are put on the water; and Sifri expounds this from a verse!

Answer (Tosfos Yeshanim DH Rashi): Perhaps mid'Rabanan Melachah disqualifies even after the ashes were put on.

Question: Sprinkling on an animal is like carrying Mei Chatas not for the need of Haza'ah, it makes the person Tamei mid'Oraisa. How can he sprinkle on the person afterwards? It says "V'Hizah ha'Tahor Al ha'Tamei"!

Answer #1 (Tosfos Yeshanim, ibid.): Since he took the water for the need of Haza'ah on a person, it does not Metamei.

Objection: The Mishnah says that if he intended to sprinkle on an animal and sprinkled on a person, if water remains on the Ezov he may sprinkle again!

Answer #2 (Boaz Parah 12:8 (b'Sof) DH v'Lo): We must say that the sprinkler must be Tovel before he can sprinkle a second time. Alternatively, immediately after intending to sprinkle on the animal he intended to sprinkle on the person; intent for the animal is ignored, it is as if he intended only for the person.