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Yes, that is the mother, but still havent found the father as yet

Thanks, have looked into those details now and anything else you find greatly appreciated.

The 1911 census I posted earlier is Mrs. Ball at 66 Kew Road but there is only Esther living with her.

If their mother is Elizabeth Ann Kelly formerly Williams,(from 1915 records) have you found that marriage, as it should give name of her Father?

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Marriages Mar 1913
Kelly John E Williams Stockport 8a 129
Williams Elizabeth A Kelly Stockport 8a 129

Regarding the 19111 census, there is an address shown for Mrs Ball as 66 Kew Road Birkdale, with the right names and ages of children present.
On the Childrens Home Record, dated the 2 December 1915, there is an address shown as 55 Claremont Road, Birkdale, so it looks like she had moved by then and that my Elizabeth, ie the daughter is Elizabeth Ann aged 16 in service.

The marriage information is correct and it is her mother with all the children, no fathers are registered with any current information.
Possibility that a father may have been Thomas Williams.
Going back to the Children's Home Record, filed on 2 dec 1915. It states that Esther was born on 22 Feb 1903 in Frodsham and baptised at St Aidens, Kew Road Birkdale.
Frances Noel Williams b 5 Dec 1904 at Stockport Union Workhouse.
Nellie Williams b 24 August 1906 at 2 Cross Street Northwich.
Mother is shown as Elizabeth Ann Kelly, formerly Williams aged 42. Address and father unknown.

She did not marry as Elizabeth Ann:
Marriages Mar 1920
Rimmer John Williams Ormskirk 8b 1360
Williams Elizabeth Rimmer Ormskirk 8b 1360

or die:
Deaths Mar 1961
RIMMER Elizabeth 61 Southport 10f 899

No Elizabeth Williams born in Stockport, but the 1911 might be based on incorrect assumpions from the Mackinsons.

If you just want to know the Mother's name, then any of the children's certificates will tell you that.
It might not tell you much else..........

Did all the children have unknown fathers?

That all sounds very encouraging,dates work out very well.
Thankyou all so far here for your help and encouragement tonight. I hope to hear more from you.

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1911 England Census
Frances Williams Age in 1911: 5 Estimated Birth Year: abt 1906
Relation to Head: Boarder
Gender: Female Birth Place: Stockport, Cheshire, England
Civil parish: Ainsdale County/Island: Lancashire
Country: England
Street Address: 52 Liverpool Rd Ainsdale
Occupation: School
Registration district: Ormskirk
Registration District Number: 457 Sub-registration district: Formby
ED, institution, or vessel: 9 Piece: 22849

Ellen Makinson 66
Robert Makinson 42
Lizzie Williams 11 Stockport
Frances Williams 5 Stockport
Nellie Williams 4 Stockport

Source Citation: Class: RG14; Piece: 22849.

Frances Williams on the end quarter sounds correct.

Esther with Mrs Ball, sounds correct and in the right area. We were led to believe that the three younger girls were all placed together, with Esther the eldest. All went to school in Birkdale together... St Peter,s or St Pauls?

Eveline sounds about the right age, only information is the rough age from a childrens home document from 1915, that lists all 8 children

There is no Frances Noel showing for Stockport, there is this one:
Births Dec 1904
WILLIAMS Frances Stockport 8a 56

Added:
from Cheshire free bmd
Cheshire Birth indexes for the years: 1904
Surname Forename(s) Sub-District Registers At Mother's Maiden Name Reference
WILLIAMS Frances Stockport First Stockport WILLIAMS ST1/6/8

This Eveline?
Births Mar 1895
Williams Eveline Birkenhead 8a 555

Added:
Also this one:
Births Sep 1894
Williams Eveline Birkenhead 8a 505

Do you mean they were all living together in 1911 or separately?

Is this Esther? She is with a Mrs. Ball?

BALL, Moses Head Married M 45 1866 Railway Passenger Guard Mow Cop Stafford VIEW
BALL, Adeline Mary Patrice Wife Married10 years F 35 1876 Dublin Ireland VIEW
GLEDDHILL, Violet Olive Boarder F 12 1899 School Yorks Bradford VIEW
DAVEY, Major Robert Arthur Boarder M 8 1903 Staffs Burton on Trent VIEW
WILLIAMS, Esther Boarder F 8 1903 Cheshire Stockport VIEW

RG number: RG14
Piece: 22860
Reference: RG14PN22860 RG78PN1339 RD457 SD5 ED20 SN24
Registration District: Ormskirk
Sub District: Formby
Enumeration District: 20
Parish: Southport
Address: 66 Kew Road Birkdale Southport
County: Lancashire

1911 England Census about Esther Williams
Name: Esther Williams
Age in 1911: 8
Estimated Birth Year: abt 1903
Relation to Head: Boarder
Gender: Female
Birth Place: Stockport, Cheshire, England
Civil parish: Southport
County/Island: Lancashire
Country: England
Street Address: 66 Kew Road Birkdale
Registration district: Ormskirk
Registration District Number: 457
Sub-registration district: Formby
ED, institution, or vessel: 20
Piece: 22860

Moses Ball 45
Adeline Mary Patrice Ball 35
Violet Olive Gleddhill 12
Major Robert Aurther Davey 8
Esther Williams 8

Source Citation: Class: RG14; Piece: 22860.

yes, that is correct

Is one of these Esther?
Births Jun 1903
Williams Esther Runcorn 8a 249
Williams Esther Runcorn 8a 221

Added:
Cheshire Birth indexes for the years: 1903
Surname Forename(s) Sub-District Registers At Mother's Maiden Name Reference
WILLIAMS Esther Frodsham Halton FRO/31/3

I am sat here talking with the (gr) grandaughter of Elizabeth, the other information came from National Children,s home

Really want to find her birth, so I can obtain the certificates. Her mother was also called Elizabeth, one had the middle name of Ann, but not sure which one.
The younger Elizabeth died in 1961in Southport. Married in 1920 to a John Rimmer.
Thanks