You can download the technical manual for the machine from here:

http://www.outdoordistributors.com/pdf/Murray/MURRAY-MODEL-40567X53B-LAWN-TRACTOR-%282001%29-PARTS-LIST.pdf

The wiring diagram is on Page 6. Joe, as I read it the seat switch only grounds the kill wire if there is noone on the seat, it doesn't seem to affect the starter solenoid. The only other safety switches I've found are on the Power Take-Off and the clutch-brake pedal, and both of those two kill the starter solenoid as well as the ignition.

Vanessa, if the engine sometimes does not crank, you may have a flat battery or a loose battery terminal clamp. These are more common faults than something wrong with the safety circuit. You can check whether this is the problem with a voltmeter attached to the terminal clamps (not the battery posts themselves). If the voltage drops way down - say 5 volts - when you hit the starter, that will be your problem.

If you want to test the seat switch, just disconnect either of the two yellow wires attached to it and see if the problem goes away, but at present I do not see how that could help since it does not seem to affect the starter solenoid.

If there is sufficient voltage across the battery terminal clamps when you hit the starter switch, but the engine does not crank, the next thing to check is the other two safety switches (PTO and clutch/brake pedal). I'm guessing you know that you have to disengage the PTO and push the pedal when you crank the engine, or nothing will happen. This is a safety feature that is also used on American cars with manual transmissions built in the last 40 years - it keeps people from starting the engine when the car is in gear (or the PTO is engaged, which could cause an attached piece of machinery to rotate and hurt someone). Those two safety switches are connected to the starter solenoid by orange wires, and probably the easiest way to test them is to check whether 12 volts suddenly appears at the starter solenoid's activation terminal when you try to start the tractor. If you have the PTO disengaged, sit on the seat, push the clutch fully down, when you hit the starter, you should see battery voltage on the terminal where the orange wire connects. If you don't, the fault is in the safety wiring, and will probably be a defect in either the PTO or clutch switch.

Please bring us up to date after checking these things. AND BE CAREFUL - THAT IS NOT A CAR, IT'S A FARM MACHINE. PEOPLE OFTEN GET HURT OR KILLED BY FARM MACHINES.