Hi Isobel
Have you discounted the English birth entries showing on FreeBMD. For example:
Births Sep 1901
Rogers Elizabeth S.Shields 10a 847
Births Dec 1901
ROGERS Elizabeth S. Shields 10a 839
On Scotlands People, there are only two possible birth entries between 1901-4 in Hamilton:
1901 RODGER ELIZABETH CROSS F HAMILTON /LANARK 647/00 0388
1902 ROGERS KATHLEEN ELIZAB F HAMILTON /LANARK 647/00 1150
Was either of these the one you have already discounted?
Regards.
Monica