Thank you very much, Shona.
Yes, I've seen the certificate for Joseph's daughter Emily. I have seen William's 1901 census, and I have it linked to him on my Hendry family tree on Ancestry. I am fairly sure that Joseph and William were related somehow, as a W.J. Burlington is an informant or witness on another certificate I have, which is someone from Joseph's family. I can check that later.
I had thought that
this William James Burlington may have been a son of William & Elizabeth Brown, but I viewed the scan on GRONI and he's actually Joseph's son. They are from the same town, though, so they must surely be brothers or cousins. It seems Joseph and Matilda had a number of children born in Bellshill, and that (IIRC) their Emily was one of them.
I have a copy of the 1911 census from SP, which lists William, Elizabeth and Emily, and she is confirmed as their only child. I don't think the 1911 census transcripts are up on Ancestry yet, though. That census has her born in Scotland, whereas on the 1901 census it says Ireland -- I've not been able to find anything yet. Her name looks to be 'Emmie Burlington' but the handwriting is tiny and very squiggly. The address is 109 [something] Street, but I can't read it.
There is a second Emma Burlington marriage on SP, to an Arthur Smith in Glasgow in 1941. Emma's first marriage was to James Milne in 1921, and I wasn't able to find a record of divorce on the same site Kate found the Hendry - Broadley divorce last week. Presumably if James had died then the marriage would have been an Emma Milne, so I'm querying that one. I can't view the certificate to double-check, anyway.
After 1921 it's anyone's guess; I can find no record of any Emma Burlingtons, Milnes or Smiths (with Burlington as a maiden name) dying in Scotland.
If you search SP for surname Milne, forename Burlington, there is a 1935 (?) birth, but I've run out of credits now. It's possible that is a child of Emma & James, with Burlington as a middle name.