I was reading David Copperfield at the time and was aware that my mother, like Barkis, was going out with the tide.
A Family Romance, ch. 5
Not only was it of prime importance to a woman like Dolly to have a man of her own, but that same man, if he were willing (Barkis again), would, in marrying her, confer on her a status which she had not enjoyed for many years.
Ibid., ch. 6
Barkis is a relatively minor character in David Copperfield. He marries Peggotty (David acts as a go-between, delivering the 'if willing' message), is mean with his money, and fades away - going out like the tide. That wonderful phrase is probably what got Brookner's attention: it's more than a little Brooknerian.
The second Barkis reference, in chapter 6 of A Family Romance, is more demanding. I wonder: if I hadn't read David Copperfield, or I wasn't reading it alongside A Family Romance, would I have the faintest idea what Brookner was going on about?
I find Mr Barkis 'going out with the tide',
ch. 30 of David Copperfield