From my research, I believe that her parents were Thomas Hardy Ryals (b. 1793 d. 1870) who married Mary Sampley (b. 1799 d. 1870) in North Carolina about 1816. In the 1860 Census she is shown as 10 years old and being born in TN and her parents are Thomas Ryals and Mary Ryals, both born in NC. She is still living with them in the 1870 census. (Info submitted by Ellyn)

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Frances RIALS SULLIVAN lived with Thomas Reils \ Ryals \ Rials, etc. (Abt 1793 - c 1875) Grave stone says he died "at the age of 80 years". The name is spelled many different ways. Thomas's wife was Mary "Polly" WALKER (10 Oct 1795 - 16 Nov 1896). She was the daughter. of John and Judith FANN WALKER.
While it is true that she is listed as Frances Rial, etc. with Thomas and Mary, she could not have been their daughter. Mary would have been 55 years old when Frances was born. There is also a John Rials "Jr" listed with Thomas and Mary who was b. Abt 1846. Was he Frances' brother?
Family tradition says that Frances and John were either illegitimate children of one of Thomas' daughters or were originally WALKER children raised by Thomas and Mary. The best that can be determined, they were foster children and not legitimate. Information provided by Thomas Sullivan

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United States Census, 1860
Name Francis Ryals
Event Type Census
Event Date 1860
Event Place 4th Civil District, Hickman, Tennessee, United States
Gender Female
Age 10
Race White
Birth Year (Estimated) 1850
Birthplace Tennessee

Thomas Ryals M 67 North Carolina
Mary Ryals F 61 North Carolina
George Ryals M 30 North Carolina
Mary A Ryals F 27 Tennessee
Julian Ryals F 22 Tennessee
John T Ryals M 28 Tennessee
John Ryals M 14 Tennessee
Francis Ryals F 10 Tennessee

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United States Census, 1870
Name Thomas Rial
Event Type Census
Event Year 1870
Event Place 4th Dist Hickman, Tennessee, United States
Gender Male
Age 78
Race White
Birth Year (Estimated) 1791-1792
Birthplace North Carolina

Thomas Rial M 78 North Carolina
Mary Rial F 65 North Carolina
Mary Ann Rial F 44 Tennessee
George W Rial M 48 North Carolina
Julia A Rial F 38 Tennessee
John T Rial M 34 Tennessee
John Rial M 25 Tennessee
Fancy Rial F 19 Tennessee