This may be William on the 1880 census. The middle initial "P" and the places of birth for him and his parents are correct, but there is no wife Juliet nor dau Blanche. Needs further research.

Wm. P. HAMILTON Self S Male W 40 MI Butcher OH OH
Christ. AUGUSTINE Other S Male W 28 BADEN Butcher BADEN BADEN
Levi CARTIER Other W Male W 38 CAN Butcher CAN CAN


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Source Information:
Census Place Lions City, Beaverhead, Montana
Family History Library Film 1254742
NA Film Number T9-0742
Page Number 26C

1. 2. Blanche HAMILTON   b. EST 1871, Of St. Joseph Co., Michigan.

No children.