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There are times I go out to my 1981 automatic and it won't start. No clicking. No turning over. Nothing. Electronics on the car work fine when this happens, the starter just does not engage.
Here are the facts:
1) The battery, alternator, and starter at fine - all checked
2) This never happens first thing in the morning, it is always after I have started it at least once for the day
3) If I stop the car and then immediately start the car, it fires right up. It only happens if the car has been started, stopped, and then sat for a while (1 hour or more).
4) This is sporadic - sometimes it doesn't happen at all after it has sat for an hour, but every now and then - it is deader than a doornail.
5) I installed one of those remote starters directly to the starter, and when this happens, it starts right up using this button
6) I have no security system that I know of
Follow the circuitry from the ignition switch to the neutral safety switch (or clutch-depressed safety switch) and thence to the SOL terminal of the solenoid. There are probably two or three connectors (under carpet!) which could have become corroded. Be observant for signs of melting of the plastic insulators on the connectors - if you find any problems in that circuit, that is likely the problem. If the wires and connectors look good, try jumpering across the safety switch as a test. If it then works, you've got a bad switch. Similarly, the problem could also be the ignition switch itself.
By your comment that the remote starter button works (which basically connects battery voltage to the SOL terminal), this tells you that the battery, solenoid, ground circuit, and starter are doing their job, so the problem resides in the wire which supplies the SOL terminal with the cranking signal.
An easy way to check the neutral safety switch is to push and pull on the gear shift lever while holding the key to the start position (with the brake firmly applied). If it starts while doing that the problem is probably in the neutral safety switch.
clean you battery ground cable...you might be using your shift cable a a ground...if it has {the shift cable} a chalky white substance on it would confirm this answer...meaning your ground may be the culprit....and I agree that your other components are good