1) HALACHAH: GOING TO THE MIKVAH ON TISH'AH B'AV AND YOM KIPPUR
OPINIONS: The Beraisa states that one who is obligated to immerse in a Mikvah may immerse in the usual manner even on Tish'ah b'Av and on Yom Kippur. Is a woman permitted to immerse on those days if her Tevilas Mitzvah falls at that time?
The Rishonim write that she is not permitted to immerse, even though there may be reasons to permit it.
(a) TOSFOS in Beitzah (18b, DH Kol) explains that today, our Tevilos are not considered to be "in their proper time." We do not follow the exact schedule of counting days as prescribed by the Torah; rather, we are much more stringent and we add days in order to be safe. Consequently, a woman's Tevilas Mitzvah never occurs at the Torah-prescribed time, and thus she may not immerse on Tish'ah b'Av or Yom Kippur. It was permitted to immerse on those days only because "it is a Mitzvah to perform Tevilah in its proper time" (Beitzah 18b).
(b) Tosfos (ibid.) cites RABEINU TAM who says that we rule l'Halachah that immersion at the proper time is not a Mitzvah. According to that opinion, it was never permitted to perform Tevilah on Tish'ah b'Av and Yom Kippur.
(c) The RI cited by TOSFOS in Beitzah (18b, DH Kol) explains that immersion at the proper time, even on Yom Kippur and Tish'ah b'Av, is permissible only in order to handle and to eat Taharos, but not in order to be permitted to one's husband. Today, since we do not immerse for Taharos, a woman is not permitted to immerse in order to be permitted to her husband. (The concession to immerse for Taharos was made due to a special necessity, out of fear that one might come into contact with Taharos right away and be Metamei them. Immersion for one's husband, though, is not immediately necessary, because marital relations are forbidden on Yom Kippur and Tish'ah b'Av.)
(d) TOSFOS here (DH Tovlin) cites the Gemara in Ta'anis (13a) which says in the name of Rebbi Chanina Segan ha'Kohanim that even if Tevilah in its proper time is a Mitzvah, it is not proper to perform Tevilah on Tish'ah b'Av because "it is worthwhile to lose a day of Tevilah once a year in mourning over the Beis ha'Mikdash." The Yerushalmi rules like Rebbi Chanina Segan ha'Kohanim.
HALACHAH: The SHULCHAN ARUCH (OC 554:8 and 613:12) rules that a woman may not immerse in a Mikvah on Tish'ah b'Av or Yom Kippur because of the first reason -- our Tevilos are not considered to be in their proper time.
2) "BNEI MELACHIM" AND "DAVAR SHE'EIN MISKAVEN"
QUESTION: In the Gemara, Rav says that the Halachah follows Rebbi Shimon who maintains that all Jews are considered to be "Bnei Melachim." The Gemara asks how Rav can rule in accordance with Rebbi Shimon here, while elsewhere -- with regard to Davar she'Ein Miskaven -- he rules against Rebbi Shimon and says that Davar she'Ein Miskaven is forbidden.
What is the Gemara's question? One ruling is not related to the other. Rav's ruling against Rebbi Shimon with regard to Davar she'Ein Miskaven should have no bearing on his ruling with regard to "Bnei Melachim"!
(a) The RASHBA explains that the Gemara knew from an accepted tradition that the most lenient ruling that Rebbi Shimon issued in the laws of Shabbos is the ruling that all Jews are considered to be "Bnei Melachim." Consequently, if Rav rules in accordance with that leniency, then certainly he must agree with all the other, less lenient, rulings of Rebbi Shimon.
(b) The BA'AL HA'ME'OR says that, on the contrary, it was an accepted tradition that according to Rav the Halachah does not follow Rebbi Shimon at all in the laws of Shabbos. Accordingly, the Gemara is asking how Rav can rule like Rebbi Shimon even in this single Halachah of "Bnei Melachim."
3) THE SOURCE FOR THE MELACHAH OF TYING
QUESTION: The Mishnah and Gemara discuss the Melachah of Kosher, tying a knot. RASHI (DH v'Eilu Kesharim) writes that we derive the Melachah of tying from the repairs that were done to threads that tore in the Yeri'os of the Mishkan. Rashi (DH Kach Hu Chayav) writes further that we learn the Melachah of untying (Matir) from the Chilazon trappers, who would untie their nets in order to adjust the sizes while hunting for the Chilazon.
However, the Gemara earlier (74b) concludes that the source for both tying and untying is from the Chilazon trappers! Why, then, does Rashi here cite a different source for the Melachah of tying? (TOSFOS REBBI AKIVA, Mishnayos)
(a) The TIFERES YISRAEL (Boaz, #1) suggests that according to Rashi, a person is Chayav even when he ties an unprofessional knot (see Insights to Shabbos 112:1). Therefore, the nets of the Chilazon trappers cannot be the source for the Melachah of tying, because the knots that those trappers tied were professional knots. For this reason, Rashi explains that the source for the Melachah of tying (even unprofessional knots) is from the Yeri'os that needed to be repaired.
(b) RAV YAAKOV D. HOMNICK (in NACHALAS YAKOV #38) writes that Rashi here (DH v'Elu) explains that in order to be Chayav for tying, one must tie a permanent knot. The knots tied by the Chilazon trappers in their nets were not permanent (since they were untied and retied elsewhere). Therefore, Rashi cites the torn Yeri'os as the source for the Melachah of tying.
Rav Homnick adds that in the Gemara earlier (74b), Abaye says that the source for the Melachah of tying is from the Yeri'os that tore. Rava argues and says that the source is from the nets of the Chilazon trappers, which is how the Gemara concludes. The argument there, he suggests, is based on where permanent knots were tied. Abaye maintains that the knots tied in the nets of the Chilazon trappers were not considered permanent (as we mentioned above), whereas Rava considers them to be permanent. Since the simpler case of permanent knots is Abaye's source of the torn Yeri'os, Rashi cites that opinion here. (Rava, on the other hand, prefers the less simple explanation in favor of deriving both tying and untying from a single source.)