Dear Mr. North,
Thank you for posting your request on History Hub!
We suggest that you review these Genealogy: Passenger Arrival List Research Tips and NARA’s page on Immigration Records for suggestions on how to locate immigration information for Henry W. and Mary North. There are numerous ways that individuals could enter the United States from Europe such as on a ship that arrived at various coastal ports or over land from Canada, etc. Ship passenger arrival lists were a requirement beginning in 1820, but that does not guarantee that person was recorded or that the list still exists. Many of the passenger lists have been digitized and are name-searchable online using Ancestry and FamilySearch. There may be a fee for using Ancestry. Instead, you may wish to contact your local library. Many library systems subscribe to these sites, making them free for patrons.
We also suggest that you review the FamilySearch research wiki on Ireland Emigration and Immigration.
We hope this is helpful. Best of luck with your family research!
I have used all of the suggested links before as well as being a member of different sites. I did manage to stumble on Henry's Naturalization card that shows his immigration to NY in July 1865.I looked at all NY entries for July 1865 and did not locate them. This could be just a case of a lost or destroyed record but at least now I have something more concrete.