Passive clauses without auxiliaries
Sentences such as I came across a letter that was typed by her secretary, can be "reduced" by removing that and the auxiliary verb, yielding I came across a letter typed by her secretary. In this particular case, the by phrase, or some other modifying phrase such as that morning or in red ink appears to be obligatory. You can't simply say, *I came across a letter typed, although you can say I came across a typed letter.
In other cases, though, this seems to be OK; consider All the students arrested were from King's College.
In what cases is a dependent required to make the clause grammatical? Can anyone point me to a discussion of this in a reliable reference grammar, preferably the Cambridge Grammar of the English Language? I see it mentioned there on p. 78, but this particular issue is not discussed.
Solution 1:
Note that this pattern also exists for the verb to be: the man who is tall becomes the tall man whereas the man who is in the room becomes the man in the room.
I think the answer to your question lies in the rules for adjectives which follow the nouns that they modify. These reduced adjective clauses are, for all purposes, adjectives themselves and when they are reduced to a single word, they are positioned as a normal adjective, that is before the noun. However they follow the noun when they have more than one word after being reduced, just as with other multiple word adjectives in English (except hyphenated adjectives).
For example, you can't say He has a black as the night soul but you can say He has a soul black as the night. The adjective phrase black as the night follows the noun soul because it is a phrase not a single word.
Solution 2:
In both of your sentences, what you have is the passive participle typed, which syntactically functions as a noun modifier. When typed is being used alone, it follows the normal rules for modifier placement and precedes the noun:
I found a typed letter.
However, as a passive participle typed is able to take an adverbial complement that specifies the agent. When this occurs, the entire phrase must then follow the noun.
I found a letter typed by the secretary.
This rule is not unique to passive participles: in general, English requires single-word modifiers to precede a noun, but multi-word modifiers to follow the noun. (The exact rules are fairly complex, but the preceding rule of thumb covers the most common cases.) You can see the same thing with active participles ending in -ing and ordinary adjectives.
I saw a red ball.
I saw a ball red as the rising sun.
I met a dying man.
I met a man dying of AIDS.
I don't have a copy of the CGEL to refer to, but if you look up the section on adjective and adjective phrases I'm sure you'll find a discussion of adjectival word order.