How do I find a hidden profile in a Time Machine backup of Thunderbirdto an external HD

I opened TM and moved to a date for a backup and then opened the path. It had a tilde ~ at the beginning and had thunderbird as one of the steps. I clicked and got a spinning star that would not stop. Started again. I removed both tilde and thunderbird, put in my user name and clicked go. It pinged and highlighted Music and I opened but there were no songs listed as I have none but nothing for thunderbird. I reinserted thunderbird and clicked go but same results as before with music highlighted. I tried a variety of other things, all with no success. Hope you might have another idea or tell me if I am doing this incorrectly in one way or another.

I have tried many different ways to get Google information about the location, some sounded very promising, and I have triued all results but have not found one that has worked for me to find the mail location or the profile location. It is very upsettiing to have lost many years of saved emails to have lost them just to update Thunderbird. :(

Looked and did not work for me. Sent message but only UH can use the help offered. I still am having the problem I have tried to d/l a new Thunderbird and it says can not open because profile is missing. Went to TM backup and installed old TB from there but same message - profile missing. I open TB on other machines but do not have the older saved emails there as updated those TB's also and lost all saved emails. One is a new laptop with new TB installed but it also did not retrieve my previously saved emails HELLLLLLLL P!!!!!!!!!!