This looks like a good candidate for William in 1871:
1871
RG10/1430 17 24
Towersey, Buckinghamshire
3 Horse Shoes Inn
Caleb North, head, 48, Publican, b. Towersey
Selina, wife, 48, b. Oxon Thame
Richard, son, unm, 23, b. Towersey
William, son, unm, 21, b. do.
Hannah, dau, 20, b. do.
Martha, twindau, 20, b. do.