The details

First, it was correctly assumed that the elongated oval (cartouche) contained a royal name.

The only cartouche that appeared 6 times was assumed to be that of Ptolemy, since the Greek section showed that the inscription was about a Ptolemy.

It was also assumed that the characters sound out the Greek form of the name "Ptolemy".

On the Bankes obelisk, the Greek section mentions 2 royal names: Ptolemy and Cleopatra. In the hieroglyphic section two cartouches occured close together.

One of them was nearly identical to the cartouche on the Rosetta Stone:

Rosetta Stone
Bankes Obelisk

Then the other cartouche from the Bankes obelisk was thought to contain the name of Cleopatra.

Now consider Ptolemy and Cleopatra together:

A. Ptolemy
B. Cleopatra

A. Ptolemy
B. Cleopatra

• $A$₁ = B₅: Must be P.
• $A$₄ = B₂: Must be L.
• Then $B$₁ must be K.

Substituting the known letters into Cleopatra.

The 2 vowels E and O between L and P are probably the equivalents of $B$₃ and $B$₄, respectively.

In some forms of the Cleopatra cartouche, $B$₇ is replaced by $B$₁₀, which is also $A$₂. Both are probably T.

$B$₆ and $B$₉ must be A.

Now we have

The last 2 signs ($B$₁₀ and B₁₁) have been shown to be an honorific for goddesses, queens and princesses.

That leaves $B$₈ = R.

Now substitute the known letters into the cartouche of Ptolemy.

There are items that don't correspond to part of the name (which in Greek is Ptolemaios).

But there are other forms of Ptolemy on the Rosetta Stone.

Compare:

Rosetta Stone
Rosetta Stone
Bankes Obelisk

The other stuff ( ) must be the "royal epithets" which in the Greek text read "living for ever, beloved of Ptah".

The last sign in the simplest form, , must be the S of Ptolemaios.

The remaining signs ( and ) must translate to M and an I-like vowel.