Follow a century with another (3,2)
I believe the answer is:
act on
'follow' is the definition.
(as in acting on or following advice)
'a century with another' is the wordplay.
'century' becomes 'C' (common abbreviation).
'with another' becomes 'ton' (I can't explain this - if you can you should give a lot more credence to this answer).
'a'+'c'+'ton'='ACT ON'
(Other definitions for act on that I've seen before include "Respond in accordance with -- London district (3,2;5)" , "-- Bell, pen name of Anne Bronte; W. London area" , "Carry out (instruction)" , "Area in West London" , "Part of the London borough of Ealing -- British historian, d. 1902" .)