Elision

The word comes from Latin e-lidere, to knock out.
We omit (knock out) a vowel or even a syllable.

For fluency of speech we say:
I don't, won't, shan't.
he doesn't, can't, wouldn't dare
Who's there?
We drop letters and, to show where those letters were supposed to be, we put an apostrophe in writing.

The same shortening takes place in other languages, French for instance, and ancient Greek of course.
δι' ὀλίγου for διὰ ὀλίγου, soon (chapters 1b, 2b...)
ἀλλ' ἀεί instead of ἀλλὰ ἀεί, but always (chapter 1b, Ο ΚΛΗΡΟΣ)
ἀλλ' ἤδη instead of ἀλλὰ ἤδη, but already (chapter 2b)
ἆρ' οὐκ ἐθέλετε αὐτὴν θεωρεῖν; for ἆρα οὐκ ...., don't you want to watch it (τὴν ἑορτήν; the festival, in 4b ΑΙ ΓΥΝΑΙΚΕΣ ΤΟΥΣ ΑΝΔΡΑΣ ΠΕΙΘΟΥΣΙΝ)
ἐν ᾧ δ' ἄπεισιν ἥ τε Μυρρίνη καὶ ἡ Μέλιττα...for ἐν ᾧ δὲ ἄπεισιν, while Myrrhine and Melitta are away (absent)...

But Greek, unlike English, does not use apostrophes in the middle of words:
I pick up, lift: ἀνα + αἴρω becomes ἀναίρω, no apostrophe
I'm away, absent ἀπο + εἰμί becomes ἄπειμι, no apostrophe

  What happens here?:  

after them: μετὰ αὐτούς becomes μετ' αὐτούς
[running] after horses: μετὰ  ἵπους becomes μεθ'  ἵππους
And therefore: 'with horses' will be μεθ'  ἵππων.
 
The τ of the first word combines
with the h sound of the second and
changes therefore to θ
 
κατὰ ὁράω becomes καθ' ὁρῶ, which, as a compound verb
will be spelt: καθορῶ

He gets up onto a horse: ἐπὶ ἵππον ἁναβαίνει becomes ἐφ' ἵππον .
Why?
The π of the first word combines
with the h sound of the second and
changes therefore to φ
 
So what happens in the following phrase if we use elision for the sake of more fluent reading?
  1. Page 76 (old edition), 104 (new edition):
    I see a man not under a horse, but under a sheep
    ἄνδρα τινὰ ὁρῶ οὐχ ὑπὸ ἵππῳ, ἀλλὰ ὑπὸ οἰί.
  2. They fight from horses (=horseback)
    μάχονται (for new verb-ending cf. βούλονται they want) ἀπὸ ἵππων.
  3. I see girls with waterjars
    ὁρῶ κόρας μετὰ ὑδριῶν.