See the New England Ball Project website for more information about this line. I have only included family members for whom I can find primary sources.
Sources
Hollis, Hillsborough County, New Hampshire, Town records, 2: 17 (image 19), Eleazer Ball Jr. birth (1770); Town Clerk's Office, Hollis; digital images, "New Hampshire, Town Clerk, Vital and Town Records, 1636-1947."
"New Hampshire Birth Records, Early to 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:FLLQ-BHL : 23 February 2021), Eleazer Ball in entry for Tamson Ball, 15 Jun 1791; citing Groton, Grafton, New Hampshire, United States, Bureau of Vital Records and Health Statistics, Concord; FHL microfilm 1,000,486.
"New Hampshire Birth Records, Early to 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:FLLQ-YPV : 23 February 2021), Eleazer Ball in entry for Phebe Ball, 12 Nov 1792; citing Groton, Grafton, New Hampshire, United States, Bureau of Vital Records and Health Statistics, Concord; FHL microfilm 1,000,486.
"New Hampshire Births and Christenings, 1714-1904", database, FamilySearch (https://familysearch.org/ark:/61903/1:1:FDJ8-9PT : 18 January 2020), Eleazer Ball in entry for James Bradish Ball, 1796.
"New Hampshire Birth Records, Early to 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:FLLQ-TTG : 23 February 2021), Eleazer Ball in entry for James Bradish Or Bradich Ball, 31 Dec 1796; citing Groton, Grafton, New Hampshire, United States, Bureau of Vital Records and Health Statistics, Concord; FHL microfilm 1,000,486.
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DNA Connections
It may be possible to confirm family relationships with Eleazer by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA test-takers in his direct paternal line.
Mitochondrial DNA test-takers in the direct maternal line: