Dennis King  wrote:

> These are, whether correct or not, the readings I get.  But I'm in the
> dark, as usual, about who these characters are and what they are supposed to
> have done,

LRF: Well I can throw a bit of light on this one, anyway, because Be Bind or
Bebhionn was Brian's mother, so at least something makes total sense.
 Keating makes sure to give us Brian's pedigree on both the father's and the
mother's side (Chapter XXV), and she figures in the Ban-shenchus as well
(see Margaret Dobbs, in Revue Celtique XLVII 1930, p 338).  Lovely parallel
you draw, Helen, with the reference to Mary ending the previous line, a nice
poetic device.
I have not found anything on Eochaid mac Ardgoil at all, alas.
Murchad mac Brain is another totally anachronistic character it seems: there
was at any rate a king of Leinster by that name in 732 who was involved in
the Battle of Allan, and gets a lot of coverage on that account (look in
CELT under Cath Almaine. We had a Murchad mac Broin figuring prominently in
the early sections of the poem, now we have Murchad mac Brain, and a third
Murchad, Murchad son of Brian (BB himself) is about to show up in our last
set of verses.  I suspect that this is not accidental.  Brian Boru's son
Murchad became something of a folklore character and figures as something of
a heroic hothead.  You may remember that our Murchad mac Brain who did or
did not fight TOC was depicted in just that light.

So far we have then,

Eochaid mac Ardgoil na n-each,

Murchad mac Brain do c[h]rech magh,

tuccais i ngeimil Brīain Breg

mac Bē binn dar fled samh.

 44.Eochaid mac Ardgoil of the horses,

Murchad mac Brain who raided plain

you put in the fetters of Brian of Brega

Bé’s sweet son, whose festival is summer.

It looks kind of like the poet is making out that TOC went out and brought
in all of Brian's foes and others to him at his court in Kincora, i.e. that
TOC was Brian's hit man who made all Ireland bow down to the great ruler.
 Does that sound reasonable?  It would give an explanation for the overall
motive for the poem.  Way back when, you might remember I mentioned
Katherine Simms' suggestion that the poem was written for the court of a
fourteenth century TOC who was involved in the Bruce Wars.  If the poet was
painting TOC I as having been the scourge of the land on behalf of a great
king and ruler, it might serve as a model for TOC II to go out and conquer
on behalf of the new aspiring king, Edward Bruce.  These are just
suggestions, I'm grasping at shadows in the mist here.

Thanks Dennis, Helen, and thanks Liz for the paradigms,