Creating a SSH tunnel passing through a 'bridge' machine

You should use ssh -L option to do this

You can read this: http://www.revsys.com/writings/quicktips/ssh-tunnel.html

It is actually quite simple. It is just a matter of a single command:

ssh -N -L localhost:3306:DATABASE_MACHINE:3306 BRIDGE_MACHINE_USER@BRIDGE_MACHINE