Surveying and Levelling. 163 



v., answering to the mean latitude or J (/ -f- V), the sum will be the 

 log", of an arc on the meridian in seconds, m", to be added to the latitude 

 /', if approaching the pole, but subtracted from / if receding from it, 

 the suni or difference will give \, the latitude of the foot of the perpen- 

 dicular upon the given meridian from the point in that required. 



3. To the log. of a add the log. sine m, the azimuth, the log. P an- 

 swering to X, the sum will be the log p", the perpendicular arc in 

 seconds. 



4. To the log. sine x add the log. cosine p", the sum will be the log. 

 sine of the true latitude required.* 



5. To the log. tangent p" add the log. secant x, the sum will be the 

 log. tangent u, the difference of longitude, which properly applied to 

 the longitude of the place of observation, will give the longitude of the 

 point required. 



6. To the log. tangent of u add the log. sine i (^ + l'^, the log. 

 secant ^ (J, — ^'),t the sum will be the log. tangent c, the conveq^nce 

 of the meridians of the given and required points, which, added to the 

 azimuth m', at the latitude nearest the equator, will give m the azimuth 

 as the latitude farthest from it, and vice versa. 



7. To the log. O, answering to the middle latitude and given azimuth 

 a from the tables I., IV., or V., add the log. of the given distance a, the 

 sum will be the log. of the intercepted arc in seconds, which measures 

 the angle between the verticals of the given points. 



8. To the log. a add the correction m h, answering to tlie given height, h, 

 of the place of observation from table II., and from it subtract pa^, 

 corresponding to a, the sum will be the log. of the chord K, passing 

 through the point of observation to the point observed. 



The number S is the difference of the log. secant of half the angle of 

 the verticals and log. pa'^, or, with the proper signs, it is, log. sec 4 t> — 

 log. p a% which simplifies the computation of heights. 



9. Since the effect of refraction is taken at 0.08 of the intercepted arc 

 a, denoted by n, this must be combined in the calculation of heights with 

 the other parts of the operation. 



Galling v the angle of the verticals, then 4 (2w — 1)> =-—0.42 u is the 

 correction to be applied to the observed zenith distance ^, to get ^„ the 

 corrected zenith distance. 



10. To the log. cotangent ^, add the log. K, the sum will be the log. of 

 the elevation of one place above the otlier. 



• Since the difference between x and the true latitude, / or /', as the case may 

 be (expressed by the arc x/ in the figure), must always be a small quantity, a 

 special table will readily give this reduction without a logarithmic calcu- 

 lation. 



i* As ^ {I — /') must always be a small quantity, its secant cannot differ much 

 from radius ; it may, therefore, generally be omitted as in formulae (8) and (9). 



l2 



