EHODA.] METHODS FOR DETERMINING ELEVATIONS. 489 



however, that these observations were not taken with the object in view 

 of making such a system of trigonometric levels. Moreover, the instru- 

 ment used read only to minutes of arc. Supposing an error of a min- 

 ute in a reading, which is not at all uncommon, the resulting error in 

 the difference of level of two peaks from a single observation will be 

 15.3 feet for a distance of ten miles and 23 feet for a distance of fifteen 

 miles. If, as is sometinies the case, the error be more than one minute, 

 the error in the elevation will be still greater. Another large and 

 uncertain element in the problem is refraction, which in the high mount- 

 ains is so changeable as to add much to the uncertainty of the results. 

 In many cases the observations were taken during storms, and often 

 the peaks were sighted through breaks in the clouds, making the refrac- 

 tion still more uncertain. 



From each station angles of elevation or depression were taken to the 

 surrounding peaks and especially to previous stations. Had the fore 

 sights and back sights between the several stations been simultaneous the 

 error of the refraction correction would have been very nearly neutralized, 

 but these two sets of observations were never taken at the same time, and 

 in only one case on the same day. From each of two stations I always 

 succeeded in fiuding some peaks which had been sighted from both. 

 With this material on hand the distances were obtained from Mr. Wilson's 

 plot of his secondary triangulation, which will not probably involve, in 

 any case used, a greater error than five-hundredths of a mile, which in- 

 cludes the error due to shrinkage of paper, as these distances were all 

 hastily taken off from the map with a scale. Having then the horizontal 

 distance between the two stations and the angle of elevation or depres- 

 sion from one to the other, of course the difference of level can be deter- 

 mined. But, on account of the errors which have crept into these angles 

 from the causes above mentioned, one determination of the differ- 

 ence of level is not sufQcient. The back sight is then calculated, and 

 brings a different result. For a still finer approximation, wherever ver- 

 tical angles had been taken from the two stations to the same point, 

 the height of that above and below each station was calculated. From 

 this, another determination of the difference in the height of the two 

 stations was determined. Then the height of another unvisited point 

 was calculated, and so on for all the near points sighted from both sta- 

 tions. Each point gives one determination of the difference of the two 

 stations. In some cases it will be found that one result is far out from 

 the rest. This may be due to the fact that sights to different points, 

 which have received, by mistake, the same number in the notes, have 

 been used. Such cases are thrown out, rfnd a mean of the rest assumed 

 as the true difference of level. It was found that, on account of errors 

 of refraction and imperfections of the instrument, sights over fifteen 

 miles in length could not be depended on at all. In the following calcu- 

 lations no sights of that length were used, and in fact very few over ten 

 miles have been used. 



In making the calculation, the following formula was used, taken from 



dh = 0.00000485 K A ± 0.000000667 K^ 



In which <?/iis the difference of level of the two points, K the horizontal 

 distance in yards, and A the number of seconds in the vertical angle 

 used. In this formula are contained corrections for both curvature and 

 refraction, the latter element being assumed equal to 0.078 of the curv- 

 ature. On exaoiining the notes carefully it was found that there were 

 sights to many hundreds of different peaks, and it became a difficult prob- 

 lem to utilize all this material, and at the same time do it according to a sys- 



