Putting for consideration these censuses:
1861 Living in Birmingham
Enoch Dance 28
Elizabeth Dance 27 Birmingham
Elizabeth E Dance 6
Arthur Dance 4 born Harborne, Staffs
Enoch Dance 2
Sidney Dance 10/12
Emma Westbury 15
1901 living in Erdington Birmingham
Elizabeth Dance 67 Birmingham Head
Elizabeth E Dance 46
Arthur Dance 44 Birmingham occupation : Mining Engineer-Transvaal
Enoch Dance 42 button makers manager
Sidney Dance 40 widower commercial traveller
Arthur G Dance 3 grandson Ancestry says Pretoria France I say Pretoria Transvaal British Subject
Genever Dauncey 20
Now in 1911 Arthur and Emily have been married no children, and he does not look like a 'mining engineer', but we have not found them elsewhere so just as an outside chance. Also there could be mis-understandings and wrong entries and Sidney could be the worker in the Transvaal and he is the father of Arthur G.
Elizabeth Dance is in censuses etc etc and dies in 1922, probate to spinster daughter Elizabeth.
Added
Arthur G looks to be Arthur Galpin Dance from later ER's, still with grandmother, uncle (?), etc, but cannot see travel for him or others.