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The observations for latitude were made under very unfavorable circumstances. Undulations in the atmosphere, heavy winds, and great changes in temperature constantly affected the instrument.
It is customary in this office to select for latitude only thirty-five pairs of stars, which have to be observed on five different nights. But the observer is dependent upon the weather, and is frequently disappointed, although after waiting I preferred to select a greater number of pairs, believing that the final result of a latitude is better when depending upon various star-places, giving the probability that the errors resulting from the declination-places of the stars will more nearly compensate each other, and that this part of the probable error of the final result will come within that resulting from observation alone.
The mean latitude is obtained by taking the mean of all single results. For the different days the mean latitude is found to be as follows:
showing a great difference between the first and second day, arising from the disturbed condition of the air.
where v is the difference between the mean results and the single results and n the number of observations; therefore the probable error of the mean result is,
If it is proper to place all the observations in the final result with the same weight (as in determining the longitude of a station from different nights' work) the formula should be used in this way; but in determining the latitude of a station, every single result obtained also depends upon the places of the stars forming the different pairs. It is certainly wrong to determine the probable error of the latitude-result by this formula, (it would give, for latitude of Colorado Springs, a probable error less than oʻʼ.01,)
though it is frequently done. If nearly the same number of stars are observed every night under the same conditions, I should prefer to determine the probable error of the final result after the manner of Mr. John H. Clark.
Probable error of one pair of stars, including constant errors
of zenith-telescope observations
Probable error of one observation .
Number of pairs used at the station
then probable error of the final result,
The formula shows that if the stars used are not very good, it is then better to select a larger number of pairs of stars, giving the probability that the final result will be more independent of the declinations.
From all the observations of pairs of stars observed on three or more nights, I find the probable error of one observation—
and that of the final result,
Taking the value for &,, found by Maj. C. B. Comstock, of the United States Lake Survey, for stars taken from Professor Safford's Catalogue for 981 Stars, ±0.53, the probable uncertainty of the final result will be, o".082.
Resulting Astronomical Co-ordinates for the Astronomical Monument at Colorado Springs, Colorado Territory, using, for the longitude of Washington and Salt Lake, the same data as in Clark's report.