Thanks for all help and comments.
Not much time lately so slow progress.

I have adjusted the toe in several times. I made a laser measuring device as per the other threads. My mark 2 version I think is fairly accurate and have adjusted the toe in to 0.05 degrees on each wheel. (Slightly beyond the 0.2+/-0.2 which I believe is specified) This requires 180 degree turn of each track rod end.

I have also taken the brand new chinese tyres off the back and put back on the almost worn out Perellies. The man changing the tyres advised me the chinese ones were not suitable for jags.

I will now run around for a few 100 miles and see how it goes. The car is certainly a lot better but not right yet. My wife was sitting in the back and after a few big bumps was quite concerned that I hadn't tightened up the wheel nuts!

 

That is immaterial when doing it by properly squaring off the string "frame" based on reference points other than the tires.

 

The trick with strings is to ensure they are parallel. Use two pieces of wood (or similar) EXACTLY the same length and place one in front and one behind the car. Stretch a string along each side of the to pass by the ends of the wood. The wood will hold the strings exactly parallel. Measure from the wheel hubs to the string on both sides of the car. Move the rear piece of wood from side to side until the parallel strings are an equal distance from the rear hubs each side. Repeat moving the front piece of wood and measuring from the front hubs. Check and recheck front and back. Once you are happy you can measure from the strings to your wheel rims and calculate the toe in.

A laser is little more than an electronic piece of string. Projecting a laser forward from the front wheel is all very well but how can you tell if your laser is exactly parallel to the wheel? I overcame this problem by mounting the laser in a 100mm deep channel section of aluminium that I used toes up. I could then use the channel toes up both sides of the car and any inaccuracy in the laser mounting would simply cancel itself out. Of course I am relying on the channel being truly parallel but I suspect it is more parallel than my aluminium rims are true!

 

Any idea of what your starting values were?

 

If you look at the Gunson device mentioned above, it relies on a mirror. That is how it frees the device from the problem of parallelism. Quite a clever application of optical properties.

 

IT WAS THE REAR TYRES!

I have just taken the XK8 for a 230mile test drive having replaced the two new cheap tyres on the back with the two almost worn out Perellie tyres that they replaced. The car behaved beautifully and the cold sweats that broke out before any fast corner of lane change have now vanished.

In case you were wondering, the dealer I bought the car from replaced the two rear tyres as part the deal. I asked to keep the tyres he removed as they had the odd spare mm of tread. It looks like I will have to save up for 4 new tyres now!

Thank you all for all your interest and advice.

 

Hello Steve, Glad you found out the problem that was causing your symptoms.
I know that I'm replying a month too late, however you may want to add to your knowledge base by not repeating other's mistakes.

I brought a car to a shop for a front end alignment because the dealer had a coupon for the service. (not a Jag). 90 days after the "alignment" I needed new tires. They didn't do it right. When I paid for a proper alignment, again, I asked for the readings "before and after" which the shop provided. The alignment was pretty bad before they adjusted it to "close to factory spec". The point of this was that after the first mis-alignment, I couldn't tell that there was uneven tire wear. That takes time. Having just bought a car, you don't really know whether the tires are wearing evenly, all you know is that at that moment, the rubber looks even.
Next learning - I accidently set up the steering wrong on a two seater sports car (not a Jag) and had the exact experience you are describing. With spirited turning, I noticed that the car tended to oversteer and wrap itself into the turn! Much more frightening than the understeer I was expecting. When I backed off the steering, it swung in the opposite direction with a vengance! Turns out that I had dialed so much camber into the front end that when the car shifted weight to the outside of the turn, the front tire had far more camber than the rear tire and gripped and pulled harder into the turn. It was a combination of the car's geometry's roll-steering and improper camber that caused the problem.
Now the fun part. You discovered that under the same side-load, the rear tires are less grippy than the front tires. As long as you are being socially responsible and not driving with passengers, or near other people or objects, it sounds like a fun setup for drifting? But that's not advice from me. I'd be afraid that you would be traveling at high speed some day and swerve suddenly to avoid a lemming or something in the middle of the road and you'd be back to unpredictable oversteer.

Glad you figured it out.

 

Since writing this thread I have continued to tinker with my xk8 as although now safe to drive it was not good enough.
I dropped the rear suspension and diff and replaced every rubber bush. The car improved but not enough. I dropped the front sub frame. Three of the four mountings had almost let go so all replaced along with lower wishbone rubbers and all ball joints. Upper wishbone bushes and wheel bearings were done earlier. Again better but not what I expected from a jag. I bit the bullet and replaced all four shock absorbers. Cat shocks almost £300 each. Ouch. The car now rides the bumps much better. One of the old shocks off the back was much stiffer than the other or the new so I wonder if this was another problem? I think the car is ok now but will have to compare with another sometime.

 

Did you change your shock top mounts? I just did this and the ride feels much better, but it could of course be a placebo effect.

 

I put the new shocks on last week with new top mounts. I could not see any deterioration of the old top mounts and doubt their renewal would have made any difference. When I took the old shocks off I tried compressing them by hand. One was noticeably stiffer than the other. By comparison with the new ones I think the stiff one was faulty

 

Reading the initial tracking issues the member was having at the beginning of this thread reminded me of many (many) years ago, when I once mixed radial tires with bias-ply tires on my car (not a Jag; and I was very young, very poor, and scrounged what tires I could). S.williams815 description of the car's behavior very much mirrored my experience, and brought back that memory that had been long forgotten.

The mixture of radials and bias-ply tires made any movement of the steering wheel unpredictably exaggerated, and was scary to drive even at below-legal speeds.

After driving only a few hair-raising miles from my parent's house, I very carefully returned home, removed the offending tires, and never tried that again.

In more recent times, I too bought a used vehicle that the dealer had put a few fairly new Chinese tires on. They tracked ok, but had precious little wet traction, and any attempt to move from a stop on damp pavement easily induced unintentional wheel spin (on a Honda CRV!) I replaced them within a few weeks and slept much better at night afterwards.

An important lesson for anyone on this forum that doesn't already know: CHINESE TIRES ARE DANGEROUS, CHEAP JUNK! Friends don't let friends or family members drive on Chinese tires. His description makes me wonder if the tires he bought had obsolete bias-ply construction, and were 'mislabeled' as radials perhaps?

Personally, I wouldn't be even slightly surprised if that were the case, especially given s.williams815's clear description of the car's crazy behavior.

My youngest son, who now enjoys the trouble-free living with my old CRV, recently went shopping for tires and I warned him sternly not to fall for the cheap Chinese tires; he ended up buying name-brand used tires at a good price, and we're both happy with them.