At Polheim, during a period of twenty-four hours (December 16 —­ 17), observations were taken every hour with one of the sextants.  The observations show an upper culmination altitude of 28deg. 19.2’, and a resulting lower culmination altitude of 23deg. 174’.  These combining the above two altitudes, an equal error on the same side in each will have no influence on the result.  The combination gives a latitude of 89deg. 58.6’.  That this result must be nearly correct is confirmed by the considerable displacement of the periods of culmination which is indicated by the series of observations, and which in the immediate neighbourhood of the Pole is caused by the change in the sun’s declination.  On the day of the observations this displacement amounted to thirty minutes in 89deg. 57’, forty-six minutes in 89deg. 58’, and over an hour and a half in 89deg. 59’.  The upper culmination occurred so much too late, and the lower culmination so much too early.  The interval between these two periods was thus diminished by double the amount of the displacements given.  Now the series of observations shows that the interval between the upper and the lower culmination amounted at the most to eleven hours; the displacement of the periods of culmination was thus at least half an hour.  It results that Polheim must lie south of 89deg. 57’, while at the same time we may assume that it cannot lie south of 89deg. 59’.  The moments of culmination could, of course, only be determined very approximately, and in the same way the observations as a whole are unserviceable for the determination of longitude.  It may, however, be stated with some certainty that the longitude must be between 30deg. and 75deg.  E. The latitude, as already mentioned, is between 89deg. 57’ and 89deg. 59’, and the probable position of Polheim may be given roughly as lat. 89deg. 58.5’ S., and long. 60deg.  E.

On the accompanying sketch-chart the letters abcd indicate the field within which the first Polar station must lie; abcd is the field which is thereby assigned to Polheim; efgh the field within which Polheim must lie according to the observations taken on the spot itself; P the probable position of Polheim, and L the resulting position of the first Polar station.  The position thus assigned to the latter agrees as well as could be expected with the average result of the observations of December 15.  According to this, Polheim would be assumed to lie one and a half geographical miles, or barely three kilometres, from the South Pole, and certainly not so much as six kilometres from it.

From your verbal statement I learn that Helmer Hanssen and Bjaaland walked four geographical miles from Polheim in the direction taken to be south on the basis of the observations.  On the chart the letters efgh give the field within which the termination of their line of route must lie.  It will be seen from this that they passed the South Pole at a distance which, on the one hand, can hardly have been so great as two and a half kilometres, and on the other, hardly so great as two kilometres; that, if the assumed position of Polheim be correct, they passed the actual Pole at a distance of between 400 and 600 metres; and that it is very probable that they passed the actual Pole at a distance of a few hundred metres, perhaps even less.