--- Here's the text translated from Danish to English with Google translate:
"Universal power supply for many types of equipment. stabilized
This universal power adapter can be used as power supply if you have lost the original power supply. Stabilized universal power supply with input voltage of 230V ~ 50Hz (AC).
Universal power supply with adjustable output voltage. This universal power adapter can be used for many different types of equipment. 1.5 / 3 / 4.5 / 6 / 7.5 / 9 / 12V (DC) Max. 600mA.
Includes 6 different sample sizes for this universal power supply. The variety of sample sizes make it possible to use this universal power supply for most devices. (Diameter: 2.5 / 3.5 / 5.0x2.1 /
5.5x2.5 / 3.5x1.35 / 5.5x1.5 mm)
This universal power supply is stabilized, which means that universal power supply is equipped with Vehicle Stability Assist which ensures that the universal power adapter delivers a uniform continuous voltage.
An electronically stabilized universal power supply protects the environment because it does not consume power while not being used. You are therefore helping to reduce CO2 emissions.
Universal power supply can be used for most appliances that gets power from a power supply.
It looks like your Logitech thingie needs a triple-output power supply with 5V at 500mA, 8V at 200mA, and 24V at 750mA all available at the same time, while the Wall-Wart only puts out one voltage at a time.
Plus, it has no 24V output, so it looks like it won't work that way either.
What you have there (the link) is a universal battery replacement wall wart - and it will only output one voltage at a time, will never output 24 VDC nor could it supply all those currents simultaneously. You could use three individual warts to power your device - or just get the correct replacement, or go to Jameco or some other power supply supplier and find a triple output power supply meeting your requirements.
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