'02 S-Type, in the trunk, I have a large black tube that's not connected to anything & a small white connector plug that doesn't go anywhere.

I was figuring the black tube was for battery vent, but I can't find a vent on the generic battery the previous owner or dealer threw in. The black tube goes thru a grommet in the rear panel & comes out inside the rear bumper

See pic...

 

The black tube will be for the battery vent. Here's the one on my XK8:



The OE Jaguar battery has a right angled outlet on the left which is attached to the black plastic tube to vent beneath the vehicle. Your S-Type will be the same arrangement.

The white connector will be for the Transit Relay:



This is fitted to prevent battery discharge during vehicle shipping. It is removed by the dealer during PDI (Pre-Delivery Inspection) and the lead should be tucked out of sight behind the trim.

Graham

 

I've heard about that transit connector. I'll have to tuck it away.

I hooked the black tube to what I think is the vent on this 'Duralast' battery. I believe the vent is in between the two cell sections facing to the right on it.