Hi,
Dave is right, check the vent on top to make sure it's not blocked in any way. A clogged vent probably starts most leaks. As the oil heats up it expands and has to go somewhere. The front seal is the most vulnerable if the vent is plugged.
If you find the vent needs cleaning, be *very* careful not to cause any dirt or grit to drop down inside the diff. Maybe try using a vacuum cleaner to remove any dirt around the vent.
But, if the car has sat for a long time, I would probably want to remove the back cover, look for and wipe out any sludge that's gathered on the bottom. This would also be a good opportunity to simply look over the ring gear & pinion, for any obvious broken teeth or any major concerns like that. A new rear cover gasket with a little Hylomar on each side will seal it all back up.
There has been a lot of discussion here about the correct modern oils to use. You might search the older postings. I'm using GL4 85w95 gear oil in my TR4's diff. I've heard both, that GL5 shouldn't be used in the diff, like the gearbox, because it contains brass parts. Others have said that GL5 is fine in the diff (but not in a TR gearbox).
Sounds like yours is stock (non-LSD), but a limited slip differential would need different oil.
A couple other things in and around the rear axle...
Do you find any oil in the brake drums? If so, the axle tube seals might need replacement. It's not a difficult job anyway, but is particularly easy when the body is off the frame. Just unbolt from behind the brake backing plate and the entire axle and hub slide out. The seal is an inch or two inside the axle tube. The only precaution is to handle the axle splines, in particular, carefully. And, grease the splines a little to help them slip past the new seal without damaging it's lip.
There are actually grease seals within the hubs, too, that go bad and allow oil to get onto the brake assemblies. But, these require separating the hub from the axle, which is nearly always a chore. So, while the hub is off the axle, it's nearly always a good idea to go ahead and put in new bearings.
Axle end play should be checked, espcially if new bearings are installed. It's easily adjusted by shimming under the brake backing plate. The nominal end play should be about .005", if I recall correctly (check your manual to be sure). Finally, the shim packs are equalized, same on each side as much as possible, to centralize the axles inside the diff.
Also, check drive shaft attachment, front and rear. Both tend to shake loose. Having a driveshaft come loose at speed is nasty, so it's an important thing to check. When I disassembled it, I found my TR4's driveshaft with 6 out of 8 bolts finger tight or looser!
New, high-grade bolts with proper length shanks and fitted with locking nuts are recommended. Triumph used and most vendors supply Nylock nuts. At a minimum, those nuts should *always* be replaced with new (3/8 x 24 NF). Personally, I like to use the stronger, aircraft grade all-metal locking nuts. TRF sells an upgrade kit with correct bolts and the aircraft quality nuts. If you have noticed any vibration in the car at speed, it might be a good idea to have the drive shaft balanced.
There's more in the rear end area that can be easily accessed and checked with the body off; such as rear leaf springs and their bushings, parking brake cables, brake lines, bump stops, shock links, and the shocks themselves.
Have fun with your project!
Alan Myers
San Jose, Calif.
'62 TR4 CT17602L