joseph_m:
Now I don't have a 24 volt source but I do have an 18 volt source. Also I only had 7809 and 7909 regulators not the 7815 and the 7915 in the schematic but the concept is the same. Here is the circuit working.
An issue with your circuit is that you do not have a common ground for the two voltages. Load or no-load, you will never read -9V between 7909 output and ground (as marked in your schematic) and so it is not appropriate if/when a dual supply is required.
You could achieve the equivalent of the above with a single 7809 regulator and just flip the terminals when a negative voltage is required.