The Rotherham Register Office birth indexes are available to check online at: http://www3.rotherham.gov.uk/RecordsSearch/index.aspx
There are no records 1840 to 1850, of a James Davis/Davies or Pain/e. There are however, 3 birth records for James Payne, one in 1851 and 2 in 1853.
I also have census 1841 through to 1881 for William Davis/Davies/Paine/Payne and wife Gheeney/Jenney/Jane. Too much to give full details here. You can contact me for more information at: bagnalls99@hotmail.com

Patrick