70 



wisco^n'Sin" academy sciences, arts, and letters. 



In leveling operations where great accuracy is required, there is 

 occasional need of correcting for curvature of the earth and atmo- 

 spheric refraction. The curvature amounts to about eight inches 

 per mile, and increases as the square of the distance ; the refraction 

 is dependent on the distance, and, as will appear from previous re- 

 marks, will vary with different conditions of the atmosphere. Since 

 refraction as a rule makes objects appear higher than they really 

 are, it serves to reduce the errors due to curvature, and according to 

 Bessel's coefficient of 0,0685, it amounts to a little more than one- 

 seventh of the curvature. 



As the earth is not a true sphere its diameter cannot be equal in 

 all places, yet this will make no practical difference m most opera- 

 tions, but it would be well in all cases to give the basis upon which 

 tables are computed. I give the following tables for corrections for 

 curvature and refraction for a mean diameter of 7925,64:6 statute 

 miles, or 41,817,247 feet. The quantities are computed to six and 

 seven places of decimals and corrected for second differences : 



Formula C= 0,4315 e"- 



Distance c. 



Curvature h. 



Refraction/. 



h-f=C. 



100 yards 



200 yards 



300 yards 



400 yards 



500 yards 



600 yards 



700 yards 



800 yards 



900 yards 



1,000 yasds 



1.100 yards 



1,200 yards 



1,300 yards 



1,400 yards 



1,500 yards 



1,600 yards 



1,700 yards 



1 mile . . . 



13^ miles. . 



\% miles. . 



1^^ miles. . 



2. . .miles. . 



3. . .miles. . , 



4. . .miles. . , 



5. . .miles. . , 



6. . .miles. . . 



Feet. 

 0.0021 

 0.0086 

 0.0194 

 0.0344 

 0.0538 

 0.0774 

 0.1054 

 0.1376 

 . 1742 

 0.2151 

 0.2602 

 0.3097 

 0.3635 

 0.4215 

 0.4839 

 0.5506 

 0.6215 

 0.6662 

 1.0409 

 1.4989 

 2.0402 

 2.6648 

 5.9957 

 10.6591 

 16.6548 

 24.9829 



Feet. 



0.0003 



0.0012 



0.0026 



0.0047 



0.0074 



0.0106 



,0.0144 



0.0188 



0.0238 



0.0295 



0.0356 



. 0424 



0.0498 



0.0577 



0.0663 



. 0754 



0.0851 



0.0913 



0.1426 



0.2053 



. 2795 



0.3650 



0.8213 



1.4601 



2.2815 



3.2853 



Feet. 

 0.0019 

 0.0074 

 0.0167 

 0.0297 

 0.0464 

 0.0668 

 0.0909 

 0.1188 

 0.1503 

 0.1856 

 0.2146 

 . 2673 

 0.3137 

 ..3638 

 0.4176 

 0.4752 

 0.5364 

 0.5749 

 0.8983 

 1.2936 

 1.7607 

 2.2997 

 5.1744 

 9.1989 

 14.3734 

 20.6976 



