Enlistment Contracts in the Roman Army

[Robert Vermaat]

Can anybody tell me when the law was passed which forced the sons to take up the job of their fathers?

Sure.

There's a rule existing in 313 that probably goes back to Diocletian that decrees that veterans and sons of soldiers are obliged to sign up when they are physically capable.
Conscription was annual, and it was common that a tax was levied that could be used to hire troops or pay bounties to real volunteers.