Some good advice here. My bet is a bad centrifugal switch in the back end of the motor and/or a bad starting capacitor, but also make certain that the on/off switch isn't full of saw dust and works properly. The flexible part of the power cable from the saw to the motor carriage can break internally and give you periods when the motor will start and then die as you move the carriage, or not start at all. The frequent movement of this part of the power cable increases the likely hood that wires have broken inside this part of the power cable where the rest of the power cable rarely moves. Turning the power switch on and having nothing this suspected part of the power cable can sometimes be found by moving the carriage back and forth with the power on, to see if the motor starts or attempts to start in any position of the carriage movement.
Anything, including the electric outlet and panel circuit can be suspect when dealing with issues like this. I like to start somewhere in the middle and see if there is power there, if I do, then I check toward the motor, if no power there I check from there toward the circuit breaker or fuse in the electric panel. Fuses do develop resistance over their life, or can also blow out with a sudden power surge. Breakers can do the same, but it's less likely. Always remove the fuse and test it for continuity with a multi tester. If it's good it will have very low resistance. If bad, it will show have very high or no resistance. If more than one fuse in the circuit, check both of them.
Charley