The particle つ has been greatly reduced to hardly nothing in Modern Japanese, but it holds on it certain cases that this lesson will discuss.
The attributive case particle つ follows nominals to indicate the possessive and survives in few words such as 目毛・睫（まつげ） "eyelash". If you pay attention really closely to how this is actually used, you'll see that this particle seems to have a locative function as well.
1. 天津神
Heavenly gods
2. 滝つ瀬とは川の流れの激しいところだ。
A takitsuse is a place where the current of a river is violent.
3. 天津日嗣
Imperial throne
4. 天つ風
Winds of heaven
5. 秋入梅
Long fall rain
6a. 夏の末つ方
6b. 夏の終わり （Modern)
The end of summer
7. 上枝・秀つ枝 (A 雅語)
Upper branch
Word Note: This last example is read as ほつえ. The opposite is 下枝, which although is normally read as したえだ should be read as しずえ as the true antonym. The ず is ultimately the particle つ.
8. おとつい
The day before yesterday
Word Note: In more standard Japanese, the last word becomes おととい. おとつい comes from a sound change of the Classical Japanese phrase 遠つ日.
9. しかも天日のごとく若くかがやかしく、悩みと憂いが兆しかけた眉は凛々しさを加え、豊葦原中国にまた見ること叶わぬような美しい壮夫であった。
Moreover, he was a beautiful, splendid man that she could never see with the likes of Toyoashihara no Nakatsukuni, though he was young and bright like the heavenly sun in addition to the chivalry in his eyebrows which foretold sorrow and grief.
From 獅子・孔雀 by 三島由紀夫.
Word Note: 中国 is another example of this particle although it's in a name.
10. どのような禍津日も、わたくし共の恋の樹を枯らす力は有ちませぬ。
No calamitous day could have the power to cut down our tree of love.
From 獅子・孔雀 by 三島由紀夫.
The conjunctive particle つ shows two repetitive actions. It is normally only seen in set expressions.
11a. 彼は行きつ戻りつしながら待った。
11b. 彼は行き来しながら待った。（More modern)
He waited while walking to and fro.
12. 矯めつ眇めつ眺める。
To take a good look at something.
13a. 彼はとつおいつ思案する嫌いがある。
13b. 彼はあれやこれや思案する嫌いがある。（More common）
He has the tendency to ponder over this and that.
14. 夢のなかではいつも父と追いつ追われつして、親雄が殺されるか父が殺されるかした。
In Chikao's dreams it was always a cat chase with his father, and it was either him or his father being killed.
From 獅子 by 三島由紀夫.
15. 浮きつ沈みつ
Floating up and down
16. 御米は依然として、のつそつ床の中で動いていた。
Oyone was still stretching and crouching in her bed.
From 門 by 夏目漱石.
Derivation Note: のつそっつ is a contraction of のっつそっつ, which is a contraction of のりつ反りつ.
17. 組んず解れつ
Locked in a grapple
Derivation Note: The previous expression comes from a sound change of 組みつ解れつ.
The Similar ～み～み
～み can also be seen in the pattern "連用形 of verb + ～み + verb with ～ず + ～み" to show two actions or conditions were being repeated in alternation. This, too, is an archaism.
18. 降りみ降らずみ (古語)
From raining to not raining
～み can also be seen in the pattern "連用形 of verb + ～み + verb with ～ず + ～み" to show two actions or conditions were being repeated in alternation. This, too, is an archaism.
24. 降りみ降らずみ (古語)
From raining to not raining