Hello everybody,
after a tutoring session with one of my student (@rholyer it’s for you), euphonic t and apostrophe t are not the same and must not be confused (like I did during the lesson 
)
I found a very interesting article on the subject written in French here.
This was hard to translate and keep this article accurate at the same time: some sentences can be hard to understand, so I rewrite them in another way right after. There are a lot of repetition in these explanations !
So, first the euphonic T (t-on, t-en, t-il) :
Sometimes we framed the letter t with hyphens (-t-), between two vowels. This letter, that we called euphonic, does not have any grammatical fonction; it is here only to make the prononciation easier. Some author call it analogic t.
- Between a verb and its subject
We insert the euphonic t between a conjugated verb at the third person singular which is ending by an a, c or e and the subject il, elle or on which are placed after the verb. We framed this t between hyphens (-t-).
We can meet this form in:
_an interrogative sentence
Pourra-t-on le joindre au bureau cet après-midi?
(Will we be able to reach the office this afternoon ?)
Comment vainc-t-on ce parasite des vignes?
(How do we beat this vine parasite ?)
_the incise (the incise proposition is a proposition which is part of a complex sentence without having any syntactic link with the rest of this very same sentence)
In this example there is no grammatical link between the first part of the sentence before the comma, and the rest of it, after the comma:
Je suis ravie de vous rencontrer, a-t-elle répondu.
(I am glad to meet you, she said)
_the inversion
Ainsi demeura-t-elle chez ses amis.
(So she stayed with her friends)
- In some specific names and expressions
Some specific names and expressions have an euphonic t. For examples:
- Il adore les qu’en-dira-t-on et n’en rate pas un au bureau.
(He loves the what we will say and does not miss one at the office)
what we will say can be translated by rumor - Ce petit va-t-en-guerre nous attirera des ennuis!
(This little go to war will get us into trouble!)
go to war can be translated by swagger always ready to fight or
warmonger or one who advocates war - Je ferme l’œil un instant, et ne voilà-t-il pas que la pluie se met à tomber!
(I briefly doze off for a moment, and suddenly it is starting to rain)
- After a conjugated verb finishing with d
There will be no euphonic t after a conjugated verb which is finishing by the letter d, because the d is pronouced like a t when we do the “liaison”
- “Prend-elle” de la crème dans son café? → here we pronounce “Prend t-elle”
Does she take cream in her coffee ? - “Répond-il” aux exigences du poste? → here we pronounce “Répond t-il”
Does he meet the job requirements ?
Do not forget to say the sound t, but you have to not write it down !
- And now the personal pronoun t’
We shall not be confused beetween the euphonic t and the elision t of the personal pronoun te and toi. Indeed, elision means that we replace the finale vowel of a word by an apostrophe (rather than a hyphen - ) in front of a word which begin by a vowel or an mute h (@valk this one is for you!)
- Va-t’en (te) (et non : va-t-en) go away
- Ouvre-t’en (toi) open yourself
- Retourne-t’en (toi) go back
See as well S EUPHONIQUE (S’EN, S’Y ET Z EUPHONIQUE).
Cut before the t
When we have to cut a verb which is containing an euphonic t, we cut as much as possible before the t.
- Toute l’équipe se demandait en silence : « travaillera-t-il demain? »
the whole team wondered in silence: " will he work tomorrow ?"
I hope this will help you, but I have to admit that it is a very specific subject which can be hard to remember!
