Foz
Just bits of info as I haven't viewed any certs. just done some searches. There is no birth of a Jane McG(C)owan in the period you are looking at with parents Robert and Sarah. There is however this:
JANE CAIRNS Birth: 14 JUL 1862 Central District, Glasgow, Lanark, Scotland
Parents:Mother: SARAH CAIRNS
There is a simultaneous entry for the same birth in Stirling:
14 JUL 1862 Kirkintilloch, Dunbarton, Scotland
I can't remember the fine detail, perhaps someone else can, but I think this normally means that the baby was born away from the mother's usual place of residence and was therefore registered in both places.
These details, extracted birth records on IGI, normally indicate that the child was illegitimate.
In respect of Sarah, there is no marriage entry showing for her to a McGowan either on IGI or on Scotlands People post 1855. There is though a death entry showing on SP in 1882 in Anderston, Glasgow City for a Sarah McGowan, other name Cairns, born 1813.
I can also see her in the 1881 Census I think:
Sarah McGowan, Lodger, Widowed, age 60, Tailoress, b.Ireland
Francis McReddie, Head, married, age 25, Puddler, b. Glasgow
Hannah McReddie, Wife, age 24, Winder in cotton mill, b. Kirkintiloch, Stirling
Address: 46 Rose Street, Govan, Lanark
Given the birth entry in Kirkintilloch, I wondered if there was some connection between Sarah and the Hannah showing in the 1881 Census.
It would probably be worthwhile you viewing Sarah McGowan's death cert in 1882 to see whether it can provide you with any additional information.
Regards.
Monica
