Hi Margaret
Just adding the link to the previous post on the family for background here
www.rootschat.com/forum/index.php?topic=687396.0A few ideas:
You can contact Scotlands People and ask for help with daughter's initial(s) if the writing is not clear. See
www.scotlandspeople.gov.uk/content/help/index.aspx?r=1210You can also add a snippet with her name and people can try to help here with making it out with you. If you have problems with this, just let me know and I can help.
Was there an address on the death certificate for the daughter? You could try searching further on the address via electoral rolls (this have to be done in person/on your behalf as not available online).
What was David Laing Darlymple's parents' names? You might get some clues from naming pattern from their names if used in any way for the naming of daughter (there might be more than one child too).
Again, sadly a search that can only be done in person at a main genealogy centre or registrars, you can search for births using mother's maiden name for births from the very late 1920s I believe.
Monica

ADDED: Sorry, see you added David's parents' names in your post.