Mary Bermingham's (m. 1706 to Pat Dempsy) father was "Maurice of Brohal." This I take to definitely be modern Broughal, Offaly. Seems logical to assume that he was a son of William of Broholloe (Broughal), who was alive there at least into 1670.
Actually, I am quite surprised by this, but the two Bermingham pedigrees, that of Mary and of Ann, widow to the last Viscount Dempsy, don't seem to connect. Anne's one goes back to her GG grandfather Walter. Mary's goes to her 3rd G grandfather, also named Walter, but they must certainly have been different men, since they had different offspring, and, since they have the same name, the two Walters, must, at least, have had different fathers.
So, if I figure correctly, the Bermingham side could have been no closer than 5th cousins, once removed. That really surprises me. Seems quite distant.
Maybe, they were more clannish, than I gave them credit for? And the important thing was the name. Or maybe, they knew their ancestors better than I gave them credit for? And knew a common one? Or possibly there is a connection on the Dempsy side?
I thought Viscount Max had no issue. His one surviving brother, Terrence, seems to have gone overseas. I'm not sure if there could be a connection further back.
The furthest I am able to get on the side of Pat Dempsey (assuming he is the same man who took out mortgage in 1638) is to his father Charles Dempsy of Coolelan, near Rathangen.
One guy named Laurence Dempsy, lived in Cloncurry very near to Rathangen, and just a few miles from Coolelan, and had at least five sons, who were said to have served in Flanders in 1687. He was said to have descendants living in the neighborhood of Rathangen into the early 1900s. I've wondered if he could possibly be Pat's grandfather, but there may have been a lot of Dempsys around. Or he may have been too close in age to Charles. I'm not certain.