The weird gadget.
The capacitor with the relay etc in it is a potential relay/capacitor
Your motor has a start and a run winding.
The low value capacitor is the run capacitor.
The high value is the start capacitor.
The high value capacitor creates some phase shift in the start winding and increases starting torque. Bigger capacitor, more torque.
Power is always supplied to he run winding (when running)
The start winding requires a capacitor is series with it to get the motor started.
Once up to speed (the AC voltage on the start winding will be about 130VAC) the capacitor needs to be disconnected.
Sometimes a small capacitor is left connected during run.
Once running the voltage on the run winding may be in the order of 230VAC.
The voltage on the winding is approximately proportional to the motor speed.
The potential relay is used instead of a centrifugal switch.
It is important that the start winding does have a capacitor of too higher value during run as this will cook the motor.
Many refrigerators use a potential relay, because it is difficult to use centrifugal switch in a sealed unit.
A potential relay has 2 terminals used to sense the voltage.
When the voltage gets to the desired level the contacts OPEN.
Some units have 2 sense connections and 2 free terminals going to the normally closed contacts, others may have one of the contacts commoned with the sense connection.
Measuring with an ohm-meter the two wires that are shorted are the normally closed contacts. If it has 3 terminals and is not marked it is a bit difficult to identify the sense connection and the common.
Put a 150w bulb in series with what you assume is the common connection and apply 110VAC to the free terminal of the bulb and the sense connection (non-shorted connection). If the relay clicks on you have found the common connection. If it buzzes you have found the contact that is supposed to be connected to the start capacitor.
If no buzzing or clicking, all is not well. Consult an expert, or get a new one. (expert or relay)
These numbers assume 110VAC operation. If you have 220V double the voltages noted.
The bulging GREY STUFF is the capacitor internals self destruction due to excessive current density in areas of the capacitor. This is because the capacitor has been operating outside it's designed temperature/current/voltage specifications.
I have seen this often with big air conditioner capacitors, and has been traced to faulty potential relays/contacts. Often the capacitor will melt before the motor. One error in selection of the capacitors is that the small run capacitor runs at much higher than the line voltage.
The dotted line on capacitor means DC voltage. The Letter is related to the design temperature/voltage/duty cycle. Often these capacitors are mounted ON the motor which gets quite hot, which in turn heats the capacitor and exceeds it's ratings.
Super X3. 3600rpm. Sheridan 6"x24" Lathe + more. Three ways to fix things: The right way, the other way, and maybe your way, which is possibly a faster wrong way.