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Bill Lionheart you wrote: "But what did they do when there is no night?

Bill
Apologies if I’ve drawn attention to this before but see also: hinks1944southpolepaper.pdf,
which reinforces what Frank has explained. Hinks wrote this in 1944. He analyzed Bowers work which was essentially the standard RN  morning, noon, and evening sights, which might not be much different in altitude but would be in azimuth. He wasn’t using grid, which didn’t come in until about 1945. 
Hinks believed Amundsen was using his (Hinks)  ‘Pole as the assumed position’ method. Hinks had lectured on this method in the early 1900s, and it’s believed Amundsen either attended the lecture or heard about it. Apparently Scott did attend the lecture but remained unconvinced. Hinks stated he was unable to check Amundsen’s calculations fully because Norway was occupied in 1944. Hinks died in April 1945.

It appears that Amundsen took a mirror AH in addition to mercury in case the mercury froze. However, the mercury didn’t freeze, so the mirror wasn’t needed. There is a photo online claiming to be of Amundsen using a mirror AH near the Pole. The counter argument is that it was a publicity shot taken close to their base. DaveP