I am working on some assumptions here, but I seem to have some names/ dates/ places that would match up with your William.
My husband's great-grandfather was Joseph Allan Fourdyce, born 1873 in Menard Co, IL; died 1931 in Beardstown, IL. 1st wife: Ethel Viola Meads, children: James A. & Harold E. (my husband's grandfather) Second wife: Nora Bell Burke, children: Otis (died young), Robert, Elva Mae
Joseph Allan's parents were William James (born 1852 in Adams Co, IL; died 1917 in Beardstown, IL) and Mary J. McKinney.
Joseph Allan's siblings were John Wesley (b. 1876- d. 1935), Charles Edward (b. 1880), Luella (b. 1885), Estella (b. 1888), and WILLIAM N. (b. 1875).
I believe this could be your William.I have a 1920 Census with a William, wife Alice, located in Brown Co., Illinois.I see Mt. Sterling is the county seat of Brown County... which is where you say your William was married.That, and Joseph Allan was born in Menard County, where the county seat is Petersburg... plus the Beardstown connections.These seem to me to be very strong coincidences.