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Thank you for your replies. Yes, the wheelchairs with four small wheels are stable, but I can't push them, except indoors and on pavements. On sand or a gravel track they are useless. Perhaps we were adventurous taking Grandma to 2800m up mountains, but surely she should not be denied beaches or canal towpaths.
She doesn't slouch. Her lightweight chair is new, but we would willingly buy another if anyone knows of a more stable one - I've not found any with a wheelbase longer than 40cm.