The Korban Musaf during the Musaf Amidah: On the first day of Chol ha'Mo'ed one says in the Musaf Amidah "ba'Yom ha'Sheini" and "ba'Yom ha'Shelishi" ...

Reason: Since it is a Safek whether it is the second day or the third day; and so on each day of Chol ha'Mo'ed, one mentions the Safek of that day.

The Chol ha'Mo'ed Leining; Explanation #1: Rashi explains that 'In the same way as Ameimar arranged with regard to the Amidah, so too do we do with regard to Keri'as ha'Torah' ...

Explanation #1 (cont.): And Revi'i, which is added on Chol ha'Mo'ed more than on other days and it is therefore recognizable as coming in honor of the day, one Leins the S'feika de'Yoma just as the Shatz says them (in the Amidah) ...

Explanation #1 (cont.): And if it is Shabbos, one Leins them for Maftir.

Explanation #1 (concl.): And of the first three Olim, the first two Lein the two S'feikos, and the third one, the following day, in which case, Revi'i re'Leins what the first two Leined.

Question #1: According to this arrangement, the order on Hosha'ana Rabah will be different, because - if the first two Olim were to Lein the S'feika de'Yoma, Shelishi would have to Lein "ba'Yom ha'Shemini", Atzeres tih'yeh lachem" ...

Question #1 (cont.): And it would not be in order to Lein on Succos a Parshah that does not belong to Sukos

Alternative Explanation: And if Rishon Leins the day before the S'feikos, and Sheini and Shelishi Lein the S'feikos ...

Question #1: It will turn out that the order does not follow the same pattern as the other days?

Question #2: Moreover, on Chanukah when one Leins the Parshah of the 'Nesi'im', the two first Olim tend to Lein the Nasi of the day, and Shelishi of the following day ...

Question #2 (cont.): Indicating that Rishon is the main Aliyah, whereas on Sukos, it is the reverse, that the last Aliyah is the main one?

Explanation #2: It is therefore fitting to introduce the Minhag that all four Olim should Lein the S'feikos of the day ...

Explanation #2 (cont.): On the first day of Chol ha'Mo'ed, Rishon Leins "ba'Yom ha'Sheini", and Sheini, "ba'Yom ha'Shelishi"; Shelishi then re'Leins "ba'Yom ha'Sheini" and Revi'i, "ba'Yom ha'Shelishi" ...

Explanation #2 (concl.): And likewise, one Leins the Sfeika de'Yoma on each day.

Clarification: They are only equal regarding the distribution, but as far as the removal, the arranging and the burning of the bowls (of frankinsence) is concerned, they are performed by the Mishmar of that week ...

Source: For so we learned in the Tosefta (4:11) 'The Avodah of the Lechem ha'Panim is performed by the Mishmar whose turn it is that week, whereas all the Mishmaros eat it'.

Reason: Because the Gemara learns it from "Cheilek ke'Cheilek Yocheilu" - implying that the Torah includes it with regard to eating but not with regard to distribution.

The Musaf of Shabbos: Now the Mishnah says nothing about the Musaf of Shabbos, but omits it - and in fact, it is included in the other Korb'nos Tzibur.

The Musaf of Shabbos (cont.): So when the Gemara says 'To include the Par He'elam Davar shel Tzibur and the Se'irei Avodah-Zarah', it could just as well have included the Musaf of Shabbos, which the Tosefta (Ibid.) specifically says goes to the Mishmar of that week ...

Question: And the reason that it prefers to mention the Lechem ha'Panim is ...

Answer: Because "Yocheilu" implies something where the eating is the main thing, i.e. the Lechem ha'Panim, whose Mitzvah was already performed the previous Shabbos - when they arranged it on the Table, and it lacks only removal ...

Implied Question: Since, regarding the burning of the bowls ...

Answer: That comes to permit the Loaves (to be eaten [Me'ilah 9a]), and it is not an Avodah pertaining to the Loaves themselves.