Do you know the town where this William Martin was born and who his parents were?
Did he have any siblings?

William Francis Martin was born August 1852 in Ulster, possibly in one of the Counties Antrim, Derry or Galway.
He and a John Martin, probably older than William, traveled from Londonderry, County Derry, N IRE to New York, New York, USA on 25 April 1865 on the steamship Stadacona. Their passage had been paid for by someone in America who 'engaged' the McCorkle Lines and paid their fare. I have not been able to determine who that person was.

At the time they left Londonderry, their emigration sheet shows their home on 25 April 1865 as Aughill (possibly Aghahull townland), Aughnish Parish, Derry. [The spelling may be incorrect - it may be a phonetic spelling of what the clerk heard]. The record also showed a Rose Martin still living at home in Aughill, relationship unknown.
Their Post Office is listed as Ramelton.
DNA testing has shown direct matches to Martins who lived in County Antrim. That may have been William & John Martin's county of birth.
William was Naturalized as an American citizen on 18 September 1871. His record is in the Court
of Common Pleas, Philadelphia, Philadelphia County, Pennsylvania, USA.
It stated he had arrived at the Port of New York (USA) in 1865.

He married Mary F. Cronin, born in Ireland, in Philadelphia on 23 October 1879.
Mary's parents, William and Eliza Cronin were both born in Ireland.

William was enumerated in the 1870 census of Philadelphia, Pennsylvania.
The 1880 Philadelphia Census shows the birth locations of he and his parents as:
William, born Ireland, Father born Ireland, Mother born Ireland
However, the 1900 Philadelphia Census shows:
William born Ireland, Father born Scotland, Mother born Scotland

William died 8 July 1904 in Philadelphia, Pennsylvania, USA.
He is buried in the Mt. Moriah Cemetery, Philadelphia, Pennsylvania, USA.

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If you have any suggestions about how I might find records of this Martin family in Ulster, I would really appreciate it.
Thank you for the advice.

Bill Martin Denver, Colorado, USA
billmartin44@ieee.org