I would think that William Smith is the illegitimate son of Emma Smith, born when she was 18 years old. In 1861 Emma and 4 month old William are living with her parents John & Mary Smith at Church Fields, Rothwell.
In 1862 Emma married Timothy Brown.
In 1871 the Brown family are living at Wood Lane, Rothwell
Timothy 26
Emma 28 (nee Smith)
George 8
Tom 7
John 7 mths
Good luck with the Smith's......
David