My 30yo 1 series used to do it when you went over 100kph and hit a bump - the front end was like a shopping trolly wheel doing 10kph. The problem built on itself and the only way to stop it was to brake and slow right down. It reminded me a lot of the occilations you can get in the steering of some motor bikes that end up putting you on the ground.
I never found out what it was but my series 3 only being 18yo never did it so I assumed that it was all the little bits of slack in the steering/front suspension of the series 1 combined to cause the problem - I suggest all of the above but a decent steering damper should do a lot to improve things as should tightening everything up and replacing worn stuff.
REMLR 243
2007 Range Rover Sport TDV6
1977 FC 101
1976 Jaguar XJ12C
1973 Haflinger AP700
1971 Jaguar V12 E-Type Series 3 Roadster
1957 Series 1 88"
1957 Series 1 88" Station Wagon
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