Im trying my to figure out what this relay controls.

 

Thats the relay for the horns, if I remember correctly the wiring is as follows:

Green is the live for the switching circuit.
Purple (faded in your picture) and black is the earth for the switching circuit that goes to the steering column, up to the wheel and is earthed at this point when you push the horn.
The three black ones are the power in for the horns and the feeds out to each horn.

 

That’s great thank you. This is my first Jag and “I know nothing”but love this car!

 

The wiring diagram that I have from this forum doesn't even show that relay ???

 

Rather strangely the Haynes manual only shows that relay on the 240 and 340s but my 1967 MK2 has it, I can't remember whether the 1959 has it and i'm not near it to check.

This is the diagram for the 240/340 models.

 

The horn relay arrived quite late in Mk2 production. My 1963 car doesn't have a horn relay or, in fact, any relays at all. The police versions had relays (and bigger dynamos ...) and some spilled over onto non-police Mk2s that were in the production at the same time.