Surveying and Levelling. 161 



employed in various latitudes, where many of the more extended 

 special tables will not apply. I allude to such tables as those 

 given by Puissant, &c. expressly calculated for a few degrees 

 of mean latitudes. I have also, on the preceding principles, 

 calculated from the same data similar tables (IV and V), ex- 

 tending between the latitudes 50° and G0% embracing the limits 

 of the British trigonometrical survey. The logarithmic values 

 of M and P are given to every KX of latitude, in table IV', 

 being most generally wanted, as well as those of O, though 

 to every 10° of azimuth only, since they are less frequently re- 

 quired, and for any intermediate degree of azimuth, the values- 

 may be found by interpolation with sufficient precision. 



Previously to the determination of heights trigonometrically, 

 the curvilineal distance, or its chord at the level of the sea, ought 

 to be augmented for the height of the lower station, since the 

 radii passing through their summits diverge proportionally to 

 that height. This correction may be obtained by the following 

 formula, or the results derived from it, arranged in a table (II). 



Let K be the chord of the augmented arc A, at the height 

 A, derived from the arc a, at the level of the sea, then, 



LogK=loga+ — ^_— ^=loga+w/t— joa2 .... (7) 



in which M is the logarithmic modulus, and g the radius of 

 curvature of the same name as a, or in feet, as in table II. In 

 the construction of the table, I have adopted mean values from 

 the same data as before, because they are sufficiently accurate 

 for this purpose, and have assumed the mean effect of terres- 

 trial refraction, equal 0.08, or about y^- of the intercepted arc, 

 which appears to be nearly the mean value derived from obser- 

 vations made in Britain and France.* 



The third table has been calculated to correct results from 

 using, in certain computations, connected with this subject — • 

 common logarithms, instead of log. sines and tangents, in the 

 case of small arcs. This gives facility with sufficient accuracy, 

 especially, when the sm^dler classes of logarithmic tables are 



^ Conversely, this table will reduce the log. of a base measured at any 

 height h above the sea, to that level, by applying mh and pa\ with con-' 

 trary signs. 



VOL. XXVI. NO. LI. JAXUAUY 1839- ^ 



