Just to take your Gay's a little further, I have ancestors of the Gay Family in Lincolnshire, but mine were from Swayfield and Bourne.
I have a James Gay (c1816, Swayfield, Lincs) married to a Ann Maria (Unknown), they had 3 children as far as I can tell:
Nancy Maria Gay (1857), Bourne, Lincs.
James S Gay (1850), Bourne, Lincs.
Sophia S Gay (1848), Bourne, Lincs.
Nancy Maria Gay is my 2G Grandmother and she married a Thomas Moyer Teat (1857, Bourne, Lincs) in 1874.
As far as James Gay's parents, this has proved quite difficult, but I think it could be a John Gay (1799, Thurlby, Lincs) and Mary Gibson (1793, Croxton, Leicstershire). If this is correct then his parents could be Robert Gay and Eleanor (Unknown).
Hope this helps and if you find any connections then please let me know.