Part I: The Burial of the Dead
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You should think about breaking this section up into four speakers. Eliot was working with dramatic monologues. You should also think about his allusions in this section:

1) The title to THE BOOK OF COMMON PRAYER (as for burial services)
2) Allusions to Ezekiel, Ecclesiastes, Isiah
3) Allusions to WWI
4) Allusions to Dante's Inferno
5) Allusions to Tristan and Isolte
6) Walt Whitman
7) Chaucer
8) Drowning
9) Greek Mythology
10) Tarot Cards - and fate
11) Other religions

Also think about winter, spring and seasons.

Note: The speaker/poet voice is actually Tiresias (the blind prophet who is Odysseus guide in the Underworld.  He speaks in fragmented sections in this poem.

PART II: A Game of Chess

The key to Eliot is usually through his allusions. In this section there are allusions to Shakespeare: Anthony and Cleopatra, The Tempest, and Hamlet.
The Aeneid - story of Dido,
Paradise Lost, Dante's Inferno, and Ovid. Most of these allusions are connected to women.
Example: Cleopatra - a suicide over love. Dido - a suicide over love. Paradise Lost - a seduction by the Devil (or snake). Dante - lustful lovers in Hell. Ovid - a rape of a woman by her brother in-law. Hamlet - Ophelia - a suicide over love.

This section can be read as a contrast of sex and love from the viewpoint of upper and lower classes. The 1st woman, the upper class, has been compared to a female Prufrock.

The title of this section comes from an obscure play that uses chess as a metaphor for stages in seduction.

The following is from a website called, "EXPLORING THE WASTE LAND" :

"From Greek mythology. Philomela and Procne were sisters. Procne married King Tereus. Tereus raped Philomela and cut out her tongue to silence her. Philomela weaved her story into some cloth to tell her sister what happened. Procne fed their son to Tereus as punishment. The sisters fled, with Tereus in pursuit. The gods intervened, changing Philomela into a nightingale, Procne into a swallow, and Tereus into a hawk (some versions of the myth vary this.)"