I was reading this sentence:

She was moving in and out without Annabell's seeing her.

Why does Annabell's have an apostrophe s?

The gerund phrase, "seeing her," belongs to Annabell. Since Annabell is used as an possessive adjective modifying the gerund phrase, an apostrophe is necessary.

See Rule 12 in this site.

http://www.grammarbook.com/punctuation/apostro.asp

The word "seeing" is a gerund in this sentence; a gerund is a verb form being used as a noun. Annabell's is in possessive form in order to modify "seeing." If you don't have the possessive form in front of the gerund, the meaning is that "she was moving in and out without Annabell" ("Annabell" being the object of the preposition "without") -- and that's not what the sentence means. The object of the preposition "without" is "seeing" not "Annabell."

http://grammar.ccc.commnet.edu/grammar/gerunds.htm#possessive

The use of the apostrophe s ('s) in "Annabell's" indicates possession. In this sentence, it is being used to show that something belongs to or is associated with Annabell. The structure "Annabell's seeing her" suggests that Annabell did not see the person moving in and out.

Now, let's break down how to determine if a possessive apostrophe should be used:

1. Singular Possession: If the noun is singular (e.g., Annabell) and possesses something, you can add 's.

Example: Annabell's car was parked outside.

2. Plural Possession: If the noun is plural and possesses something, and it already ends with an "s," you can just add an apostrophe (').

Example: The dogs' toys were scattered across the room.

3. Singular and Plural Nouns Ending in "s": If the noun, whether singular or plural, ends with an "s," you can use either 's or just an apostrophe ('), depending on the style guide or personal preference.

Example 1 (using 's): James's book was on the table.
Example 2 (using just an apostrophe): James' book was on the table.

In the sentence you provided, the possessive form of Annabell is being used to indicate that the action of "seeing" belongs to Annabell.