This William Mathias was my gt-gt-grandfather. According to his marriage certificate his father was also William Mathias, a sailor. His mother was living with him in the 1871 census and her name was Mary Ann; she was then 57, a native of Beaumaris and her occupation appers to read as "keeper of bathing lace".
William married twice. On 28th April 1857 he married Miriam Williams in Llaneilian parish church. After she died he married someone named Margaret (maiden name not known). He ended up as lighthouse keeper at Great Ormes Head. he died at Menai Bridge on 2nd November 1902. He was survived by a son, John, and daughters Jane, Mary, Ann & Catherine.