that was my thoughts as well, my granny couldnt read or write,her sisters as you say were born in different parts of scotland and the family did seem to move location regularly. i have tried using the fathers surname and various surnames the mother claimed to have, also just a christian name search for the whole of scotland. i really did think it was a legal requirement to register births in 1909/1910 though. strangely the name kay means something to me!!