Williams father, John Murdoch was millwright and tenant of Bello Mill in Lugar, in the Parish of Auchenleck, which he  took over in 1754 and operated for James Boswell (Lord Auchenleck). 

John married Anna Bruce in 1747 and William was the third born of their seven children. 

Williams birth was baptised in the Parish of Auchenleck and he attended school in Old Cumnock until he was ten, and then he attended the Kirk Session school in Auchenleck until he was 14. He was then apprenticed to his father at Bello Mill. 

In 1760, Bello Mill became the technically most advanced working mill in Britain when John installed the first iron-toothed gearing wheels, cast at Carron's Foundry in Falkirk. 

William lived and worked at Bello Mill until 1777. It was then that he went to work for Boulton and Watt at Soho in Birmingham. 

Bello Mill is now a ruin, although the gearing is still in situ. The mill was still in use as recent as the 1960's. Sometime later, the roof was removed and since then Bello Mill has been open to the elements. 

In recent years the dam which diverted the River Lugar away from the mill, has broken and the river has changed course coming right to the edge of the ruin. This closeness of the river is taking it's toll on the riverbank with the result that the back wall of the mill has moved, exposing a massive 2 to 3 inch crack running the full height of the mill. The whole mill, including the ironwork, is in danger of being swept away by the river. 

On the river bank below Bello Mill Cottage, is the cave which William and his brothers carved out. It was in this cave that William conducted his early gas experiments whilst in his teens, sometime around the mid 1770's. 

Bello Mill Cottage is still inhabited having been continually lived in and refurbished since William's time. There is a plaque on the wall of the cottage with an inscription commemorating Williams birthplace.