ADDED: SNAP, I came to the same findings! I'll leave post as is.
The only birth details I can see for William Kirsop (from IGI) is a christening entry on 19 OCT 1817 in Medomsley, Durham. There are two further to parents George Kirsop and mother Jane:
Thomas - 23 OCT 1831 Heworth, Durham
John - 14 FEB 1836 Heworth, Durham
In terms of dates, Thomas and John could connect to that 1841 entry I posted.
There is this possibility for a marriage for the parents:
GEORGE KIRSOP and JANE HUDSPITH
Marriage: 20 APR 1815 Medomsley, Durham, England
The location for this marriage entry would tie up with the entry for the christening of a son William in 1817 in the same place.
The index for the death of William Kirsop in 1857 that you mentioned showed mother as a BROWN, so again, jury's out!
Monica
PS Kids back to school next Thursday...with a little more time and normality, I'll get round to the Library at last for you
