In which case James had a sister Elizabeth.
Hmmm.
The IGI lists several children of John Burness and Elizabeth Murray
James, baptised 3 May 1835
William, baptised 28 June 1837
Thomas, baptised 7 July 1839
Elizabeth, baptised 22 December 1841
Elizabeth, baptised 4 March 1843
all baptised in St John's Episcopal Church, Aberdeen, so it's no surprise that they're not indexed on SP.
There's information about her and some of her descendants at
https://burness.ca/g0/p587.htm#i5866In 1841, at James Street, Aberdeen
John Burness, 25, born Scotland not Aberdeenshire
Elizabeth Burness, 25, born Ireland
James, Burness, 6
William Burness, 4
Thomas Burness, 2
Margaret Burness, 50, birth county unknown
Christian Banks, 65
Isabella Gray, 60
Ann Gray, 20
all born Aberdeenshire unless otherwise indicated.
I think John Burness is in Leith in 1851
John Burns, 40, Stoker
James Burns, son, 16, rope maker
Thomas Burns, son, 13, errand boy
Margaret Burns, daughter, 2
all born Aberdeen except Margaret who was born in South Leith. The FreeCEN transcription doesn't say what John's marital status is, but FindMyPast says he is a widower.
So it looks as if Elizabeth Murray or Burness died before the start of civil registration, so no help from her death certificate.
There is a death of Elizabeth Burness in Edinburgh St Cuthbert's on 14 November 1849 which might fit.
The 1861 census lists in South Leith
John Burns, general labourer, 50, born Ireland
Wife Anne Hally, 40, born Ireland
Daughter Margaret Burns, 12, born Midlothian
three younger children and
James Callaghin, visitor, 45, general labourer, born Ireland.