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Harold Godwinson was lucky to come across the Norwegian army at Stamford Bridge divided and unprepared so fast was his approach.
Had the old Byzantine general Harald Sigurdsson been aware of the proximity of Harold's troops and met them armoured and united he could well have won.
With Wilhelm le Batard landing in the south what might then have happened?
Harald had already consolidated his position in the North and at York and he could have reached London before the news of Harold's defeat reached Wihelm and he could react to it.
If Harald delays Wilhelm might have reached London first.
I doubt if anyone knows of the weather on the east coast so we don't know if Harald's ships would have been his fastest route to London or by foot/horse.
I suggest one uses Godwinson for Harold and Sigurdsson for Harald or madness will reign. I use Wilhelm for William because that is what is written for him on the Canterbury Embroidery.