56 INSTRUCTIONS FOR MAKING PILOT BALLOON OBSERVATIONS 



cosine of the azimuth angle at the B station. That is, the corrected 

 angle is given by the equation: 



e" = e'—e cos /3, 



where e" is the corrected angle; e' , the vertical angle at the B station; 

 d, divergence angle; and /3, azimuth angle at the B station. When 

 the angle /3 is 0° or 180°, the cosine is 1 or — 1, respectively, and there- 

 fore the full amount of the divergence angle is subtracted in the 

 former case and added in the latter. When the azimuth angle is 90° 

 or 270°, the correction is zero in either case. For azimuth angles 

 between 270° and 90° (first and fourth quadrants) the cosine is plus, 

 therefore the fractional part of the divergence angle should be sub- 

 tracted, and for azimuth angles between 90° and 270° (second and 

 third quadrants) the cosine is minus, therefore the fractional part of 

 the divergence angle should be added to the vertical angle. This 

 correction does not influence the height computation to a very great 

 extent unless the divergence angle is comparatively large. 



ni . However, at greater horizontal distances and greater heights 

 the influence of the curvature of the earth becomes more pronounced 

 because the computed height of the balloon is based on the horizontal 

 plane of the station. If P/ (Fig. 18) is the determined projection 

 point of the balloon at a distance d from the A station, then, according 

 to the usual formula and the measured vertical angle e, the computed 

 height of the balloon is: 



Ai = Pi P/=c? tan e 



178. However, the real height above the surface is not ^i, but may 

 be represented by Ao P2 P2' • This height A2 can be computed very 

 nearly from the triangle A P2 P2 if the difference between d and s is 

 neglected, which is usually permissible in actual practice. 



Therefore, from the triangle A P2 P^ 



ho d 



sin (g+e/2) cos (e+0) 

 or 



, jsin (e + 0/2) . , • , • „,o ^ 



/i2==« 7 — r-^ in which sm 6/2 = ?r-f> 



cos {e-\-d) ' 2R 



and R is the radius of the earth. 



179. At a vertical angle of 18° and a height of 20 km. the correction 

 becomes approximately plus 360 meters. The determination of the 

 horizontal distance is, in such a case, so uncertain that even this great 

 height correction may be of more theoretical than practical value. 



180. Influence of optical refraction. — Since the rays of light are bent 

 in passing from a less dense to a denser medium, the balloon appears 

 imder a somewhat too great a vertical angle. The deviation, increas- 

 ing with zenith distance and height of the balloon, can be deducted 

 from the measured vertical angle. In connection with cases occurring 

 in actual practice of pilot-balloon observations, this amounts at most 

 to between one- and two-hundredths of a degree of angle and can be 

 neglected, especially because the influence of the curvature of the 

 earth acts oppositely. 



