Paul, Here goes..I will try to explain this the best I can.. I am no engineer, but am quite good at figuring some things out. I include this as one of those things. This is exactly what I did, and it worked for me. Anyone that wants to explain it better, feel free to do so.. ;D
Look at my basic circuit. You will observe that it is really two circuits in one. One to the left of "A" and "B" and one to the right of "A" and "B". If you can't comprehend this, stop now and get help.
The run circuit has more windings than shown in my drawing, but when isolating the run windings for the start winding, lets keep things simple and treat the run circuit as one winding. Getting more involved with the run windings will occur if you are changing to 220 volts, which will not affect the start winding.
The first thing I would do is mark all the wires using a wire marker kit or some other means.Get your camera out and take many close up pictures too. They will help if you get in trouble. I would remove all the wiring from in front of the terminal plate. If it's within your capabilities, I would remove the motor housing cover,(If removing this housing is not within your capabilities, stop and get some help.) Make sure to unscrew the centrifugal switch first, then the long screws with nuts come out. Although somtimes, difficult, the housing will come off with a little coaxing. Make sure you don't lose any of the bearing shims. also remove the two screws in the back of the terminal plate. You now can then get to the centrifugal switch that connects to the capacitor and looking at my diagram be able to isolate the start winding.. again, the start winding will be the one that has the capacitor and the centrifugal switch in series with it.. This circuit should be seperate and have no relation to the wires on the back of the terminal plate.There are probably four wires in the rear of the plate and these are the run windings. (see picture below of the multi colored wires connected to the back of the terminal plate.)
In some instances, as Walter mentioned the start winding wires will be connected directly to the run winding wires and will have to be isolated. When you are done you will end up with four wires, which are the two above mentioned circuits. By reversing the two start winding wires, the motor will reverse.
When putting the motor housing back on, make sure no wires are rubbing and also make sure the motor spins freely..as freely as it did before you took it apart.
Now you can try and get lucky by not taking the motor apart and just guessing, but this is how I recommend to do it.. It's your choice.
This is the best way I can explain this without having your motor in front of me.. probably the hard way, but it is the sure way..
Again, this is the exact way I figured mine out. I hope this helps a little. Maybe others can chime in for some additional support.
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I might add that the capacitor, centrifugal switch and start winding doesn't necessarily have to be in the order of my drawing to work. As long as they are in series. I have saw different variations.