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I'm not sure but I believe that the phrase:
"on the night of the first day" consists of two separate prepositional phrases.
Is it correct to have a prepositional phrase embedded in another prepositional phrase?
Yes, it is correct.
This is called embedding (or recursion) and is considered one of the universals of human language. You can say:
The dog chased the cat that caught the rat that ran out of the house that was at the end of the block with the police station that...
and you can always keep going. The rule for a prepositional phrase is that it starts with a preposition followed by a noun phrase. It turns out, as complicated as it may be, that one rule for forming a noun phrase is a noun followed by a prepositional phrase (which is a modifier of the noun just like an adjective).
PP -> P NP
NP ->
-> N PP
-> N
This allows you to embed a prepositional phrase within another prepositional phrase. PP -> 'on' NP
-> 'on' N PP
-> 'on the night' PP
-> 'on the night' P NP
-> 'on the night of' NP
-> 'on the night of' N
-> 'on the night of the first day'
(I've oversimplified the grammar considerably) which gives the parse tree:
It's a little technical but you can see that you could keep adding prepositional phrases. Or rather, whatever noun phrase you create, you can create a prepositional phrase of that and use that to modify a new noun, making a bigger PP.
There exist two competitive models to describe constructions like "on the night of the first day".
One model can be called concatenative. Under this model, "on the night" is one prepositional phrase, "of the first day" is a second prepositional phrase. Personally, I don't subscribe to this model, so I cannot tell you with certainty that this model suggests that the second phrase modifies the entirety of the first.
The other model could be called the embedded model, the recursive model, the reductive model or the nesting model. I do subscribe to this model, and Mitch has already provided a sufficiently accurate top-down explanation of how this model works. The only addition I can make is to provide the bottom-up view.
At the very bottom (or, if you prefer, at the deepest level within the recursion) is the noun "day". Both "the" and "first" directly modify this noun, giving us the nominative phrase "the first day". In turn, "the first day" is the object of the preposition "of". The prepositional phrase "of the first day" is complete and coherent.
That phrase is one direct modifier of the word "night". The other direct modifier is, once again, the definite article. This gives us the coherent nominative phrase "the night of the first day".
On our last turn, "the night of the first day" is the object of the preposition "on". This yields your original phrase, which both models accept as a complete and coherent phrase.
The effective difference between the two models is that the concatenative model does not propose that "the night of the first day" -- although on its own a coherent nominative phrase -- is a phrase that occurs in the construction "on the night of the first day". That stands not only as the difference between the models but also as the reason that I reject the concatenative explanation.
In either case, "on the night of the first day" includes two prepositional phrases. Under the concatenative model, those phrases are "on the night" and "of the first day". Under the nesting model, they are "on the night( of the first day )" and "of the first day". I prefer the nesting model because "the night of the first day" makes sense on its own, whether within or without a prepositional frame.
