rhematic

The text from which these sentences come states that the by-clauses in them are rhematic. What exactly does rhematic mean here?

(34) the battle for improvement of race relationships was both won and lost by implementing this program.
(35) Kerkovian is performing acts of mercy by relieving patients of their agonizing pain.

I have no idea what "rhematic" means, but the definition at dictionary.com was: \Rhe*mat"ic\, a. [Gr. ???? of or for a verb, fr. ???, ???, a sentence. See Rhetoric.] (Gram.) Having a verb for its base; derived from a verb; as, rhematic adjectives. --Ftzed. Hall.

And I think you mean Kevorkian.

Perhaps it means they're pertinent for those specific cases?

It means that they use the gerund form a verb, i.e. "implementing" and "relieving".