How to drive 315 coills in flip-disc display
Apart from the reed relay driver, you need a 2 position switch on each dot terminal that toggles between Ground and +24V, and another switch on the common terminal (5,9,10) that also connects to +24V or Ground. The circuit should look something like this:-



When S1 is switched to Ground and K1 is closed, any dot which is switched to +24V will be flipped to Position 1, while those that are switched to ground will not be powered. Conversely, when S1 is in the +24V position any dot which is switched to Ground will be flipped. When K1 is open no dots will be powered so they hold their current positions.           

You could use relays to provide the switching functions, but a solid state solution would be more efficient. You need one half bridge driver per dot (which has two possible output levels, high or low), plus another driver that can switch either +24V or Ground (but not both at the same time!) to the common terminal. Level shifters will be required to translate the logic level signals to +24V. 

You could make the entire driver circuit with discrete transistors, or use motor driver ICs such as the L298 (which has 4 half bridge outputs per chip). 

Five 8 bit shift registers in series would be enough to produce the 37 logic signals required.