The + cable on my 1987 Series III is getting frayed and I can feel it's warm compared to the negative cable probably because it it's internal resistance.

I would like to replace this cable with the correct gauge and type of wire. Does any know what gauge of wire came from the factory? Mine is a white cable and I am not sure if it is original or not.

White is original as I recall. Jaguar used fairly heavy guage cables...probably heavier than technically required but that's a good thing. I don't know the guage, sorry, but just take it to your local parts store and match it up as close as possible...or order a known-correct replacement from a Jag specialist.

Cheers
DD