Apparently, both ggg-grandfather Peter J. Addy (~!806-1884) and gg-grandfather Pierre "PJ" Addy (1842-1909) had brothers named William, but I've found little information about either. The elder William was reportedly born around 1800, probably in France, then moved on to Switzerland with the family. Whether he emigrated to the US is uncertain, however, at least one older and two younger younger brothers apparently did so. The younger William, reportedly Peter's oldest son and PJ's brother, was in the US because he's noted to have fought in our civil war. Any of this resonating with your history?