When you remove the batteries to change them, all the entries clear. I assume it has an internal memory battery?? If so, is it easily removable or soldered in?
It's been a long time between use of my 51 so batts were removed for months. However, I installed fresh batts and programmed a couple of freqs for a few minutes, then removed then cover with the two batts for a few minutes. Then refit the cover and found the two freqs were still programmed.
As I recall, this scanner has a cap charge which should hold the memory for a day or so.
On page two the manuals says "Memory backup keeps the channel frequencies stored in memory for up to 3 days during power loss."
As Frank said, this implies that a soldered-in capacitor provides memory power when the battery is removed.
The parts list for the PRO-51 on the Radio Shack site lists this capacitor which is almost certainly used for memory backup.
C203 10555225 CAP, .047F +80-20% 5.5V 10555225 DOUBLE LAYER C0262
Replacements can probably be found at Mouser or DigiKey. The important things are to get a voltage rating of at least 5.5 volts and to match the physical size including the lead spacing. The capacitance can be higher and that will extend the backup time without causing problems.
As a work around until you can get the cap revived (if there is such a thing) or replaced, I would just plug in an ac or dc adapter into the jack marked PWR on the side whenever you change out your batteries. That should provide power and keep the memory up till you can get another set in.
If you want specifics on the power jacks go here.