Thank you, Louisa Maud, for your excellent suggestion. There were indeed some children in St Elizabeth Home. I have re-checked them all, carefully. Fortunately, the handwriting is quite clear. Sadly, none of the children's surnames matches my relative's, or is at all similar. Interestingly, the birthplaces recorded for the children are very varied, and most are outside London.
I do know the date of my relative's death, which was in the 1980s. I should perhaps explain that, although we are not close blood relations, three generations of my family kept in touch with her, and certainly for the last 4 decades of her life, visited regularly. This relative married once, in middle age. There were no children of that marriage. Of course, there is no one left alive who would have known her in 1911, and my parents' generation are dead. The younger generation who are in touch with me had no knowledge of what this relative was doing in 1911, or in the period when she became pregnant. She is likely to have been a domestic servant, (as her sisters were) probably in the London area. Thank you for your time - much appreciated! Any further suggestions you may have will be very gratefully received!
