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.