"United States Census, 1870," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M86R-2WG : 12 April 2016), William Beach, New York, United States; citing p. 7, family 58, NARA microfilm publication M593 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 552,408.
"United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MWXY-FTL : 15 July 2017), William Beach in entry for Arthur T Irvine, 1880; citing enumeration district ED 108, sheet 15A, NARA microfilm publication T9 (Washington D.C.: National Archives and Records Administration, n.d), roll 1120; FHL microfilm 1,255,120.
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DNA Connections
It may be possible to confirm family relationships with William by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with William: