Basically, using a digital ohmmeter, you pull the compressor's relay and overload off its three terminals, then check the resistance between each of these three pins. Older relays looked a bit different than the solid state one below, but pulled off pretty much the same way.
First note the two pins that read the highest resistance. The one that remains is the 'common', to which one end of both the start and main windings connect. (The common's not always the top pin, so you'll want to do this test to be sure)
Reading from that common to each of the other two, carefully note each reading. Then measure back across the two with the highest resistance, ignoring the common. That reading should be the exact total of the two individual coil readings, because you're reading through both coils in series now.
If those two sets of readings aren't within about 1/2 ohm of each other, then one of the compressor windings is shorted, and if it runs at all, it'll run hot and usually end up short-cycling on its overload protector.
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