suzilad and allycat,
Just thought you might like to have a look at this census for 1881. ...allycat, you said you didn't know where Mary Jane Coulter was living in 1881 when she would have been 6. I will give you the same information as the census, but then I will have to make a few comments, because I believe they have made a mess of transcribing from the original census.
1881 census, 77 Greenend, Old Monkland Eastern District, Lanarkshire.
Jessie Kor Coulter Head aged 38 born 1843 Irvine, Ayrshire.
James Kennedy Son aged 15 b 1866 St Quivox, Ayrshire. Coal Miner.
Jane Kennedy [Jane Coulter] Dau aged 73 b 1808 at Tarbolton, Ayrshire.
Jessie Kennedy Daughter aged 9 b 1872 Irvine, Ayrshire. Scholar.
Mary Coulter Daughter aged 6 b 1875 Glasgow. Scholar.
Martha Coulter Daughter aged 4 b 1877 Kamannan, Stirlingshire.
Robert Coulter Son aged 2 b 1879 Old Monklands, Lanarkshire.
William Barclay Boarder. aged 38 b 1843 Ireland. Coal Miner.
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I am from Glasgow, so you may not know the area, so I will try to clarify a few things.
Wherever it says Monklands (either Old Monklands or whatever), this is old fashioned. It is Lanarkshire, just east of Glasgow they are talking about.
The address on the census is 77 Greenend, which in this case I believe is Greenend in Coatbridge, Lanarkshire (or Old Monklands).
First name on census is Jessie Kor Coulter.... I think the Kor must be short for a longer name but I don't know what the full name might be. Also, after every single name on this census, they have written "mother of Jessie Kor Coulter."
But that is wrong - because Jane Kennedy is aged 73 and born in 1808.
...I believe all the others listed are Jessie Kor Coulter's children but not Jane.
Another puzzle: some of Jessie children have the surname Coulter and some Kennedy.
And Jane Kennedy has [Jane Coulter] just after her name.
Mary Coulter appears to have been born in Glasgow, and that is fine, but Martha is born in Kamannan, Stirlingshire. That is wrong. I believe they mean Slamannan, Stirlingshire - which is near Falkirk and Camelon.
...allycat, you say that on Jane Coulter's death certificate, her place of marriage is given as being Camelon. That is in Falkirk, and Slamannan is just south of these two places.
Hope you can figure it out. But Jessie Coulter is head of the house, and they are all her children - except for Jane Kennedy, who, going by her birth date must surely be her mother. Hope I have the right family here. Do you recognise the names of any of the children? If you do, then I would think that is proof enough that it is the right family.
Brian, Glasgow.