'The X-ing of Y' vs just 'X-ing Y' : why are both 'the' and 'of' necessary together?

Morning! The below is just an hypothesis, but it sounds convincing enough to me.

A gerund is special kind of word: it is both noun and verb at once (just as a participle is both verb and adjective). In its function as a verb, it can have an object:

Augustus condemned his daughter's adultery.

By condemning Julia, he set an example for the Empire.

But it can also have an article and an of attribute like most nouns:

The public condemnation of his own daughter was part of his new policy of chastity.

His laws punishing adultery would have been hypocritical without the condemning of Julia.

Whenever the is used, the gerund is marked as a noun; that is probably why it cannot have an object then, since nouns normally can't have objects. This forces the secondary argument of the word to turn into an of attribute.

Conversely, whenever it has an object, it is marked as a verb, so that it cannot have an article. If there is no secondary argument, the article is free.

Whenever of is used, it is marked as a noun, just as with the. Even though nouns can normally exist without articles, somehow of usually forces the gerund to take the article. Apparently the pattern the + gerund + of became dominant enough to render no article + gerund + of unidiomatic. I can't really say what could have caused this, except that the lack of article may somehow mark it as a verb in this case. Perhaps some more pondering will bring inspiration.

The two sentences you have written are not exactly the same in meaning.

"There is very little that a conforming POSIX.1 application can do by catching, ignoring or masking SIGSYS"

Here catching, ignoring and masking are gerunds and they function like verbs. In other words they have a subject (a conforming POSIX.1 application) and an object (SIGSYS). It is clear that the specific application performs these acts.

"There is very little that a conforming POSIX.1 application can do by the catching, ignoring or masking of SIGSYS"

The gerunds function as nouns here. In order to attribute genitive to nouns we use 's or the preposition of, the latter being the case in this sentence. It is not clear who or what does the catching, ignoring or masking to SIGSYS since nouns can't have a subject like verb forms (unless of course there is a technical reason I am not aware of which allows everyone to understand it is the specific application which performs the above mentioned acts and nothing else). In the meantime, the has been introduced to mark this change of function in the sentence. Without the, all three gerunds behave like verb forms and of is ungrammatical.