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Thank you for your replies. Sorry should have mentioned Lambeth. My mother and my aunts only ever mentioned one brother, John,
who died in the sixties. This of course doesnt mean much as they could all have decided not to talk about any brother who died young!

I think his second name might have been Thomas, as this was his fathers and grandfathers name. Also I feel he must have been Rose and William's son as they called their other son John and not after his father.

Still hopeful!

Free bmd shows only 1 plain William Smith born Lambeth that would make him aged 1 in the 1911 census - Lambeth 1d 397 Dec 1909, but the parents don't seem to use their middle names so don't think that helps.
There are several choices including second names to fit the time frame unfortunately.

So you're looking for a Smith born in London .... no pressure !

Would Vauxhall come under Lambeth ?

If Mary is the mother of William then it would show no children born to her marriage to George Offer

Have just edited, the Smiths admit to one child born and still alive.
The Offers have been married under a year with no children

William is listed as if he was Mary's child. Wonder if there's a mistranscription with the surname

Strange that the son is listed last

And the family in 1911???????????
1911 census transcription details for: 21 Catherine St Kennington S E


National Archive Reference:
RG14PN1986 RG78PN70 RD25 SD2 ED12 SN105
Reg. District: Lambeth
Sub District: Kennington
Parish: Lambeth
Enum. District: 12
Address: 21 Catherine St Kennington S E
County: London
Name Relation Condition/Yrs married Sex Age Birth Year Occupation Where Born
SMITH, William Head Married M 25 1886 Carman Printers London Vauxhall
SMITH, Rose Wife Married2 years F 25 1886 Brewery Bottle Washer London Vauxhall
OFFER, George Boarder Married M 29 1882 Carman Omnibus London Blackfriars
OFFER, Mary Ann Boarder Married F 27 1884 Brewery Bottle Washer London Vauxhall
SMITH, William Son M 1 1910 London Vauxhall

EDIT - married 2 years, 1 child born alive, still living

Is this the marriage?

Civil Registration event: Marriage

Name: SMITH, William
Registration district: Lambeth
County: London
Year of registration: 1909
Quarter of registration: Apr-May-Jun
Spouse's last name: Not available before 1912
Volume no: 1D
Page no: 659
MarriageFinder: William Smith married one of the following people:

GEPHEART, Rose Edith<<<<<<<<<
WORKMAN, Annie Elizabeth
SKILTON, William Arthur

Location ?

I have just seen on the 1911 census that my grandparents William Thomas and Rose Smith (nee Gepheart) had a child called William in 1910.

I had not heard of this child before today, my mother and most of her sisters are now deceased.

I do not seem to be able to find a birth or death date for this child. Any help appreciated.

Thank you