506 LITTLEHALES: THE SUMNER LINE 



angle, besides the co-latitude, thus becoming known are the 

 azimuth angle and its opposite side of polar distance and the 

 hour-angle, whose opposite side is the complement of the re- 

 quired altitude. The calculation of the required altitude is 

 therefore effected by the proportionality between the sines of 

 the sides of the spherical triangle and the sines of their opposite 

 angles, thus: 



sin z: sin t: : sin p: sin Z (1) 



in which z represents the zenith distance of the observed body, 

 t, the hour-angle; p, the polar distance; and Z, the azimuth. 

 Since z = 90° — h, the complement of the altitude, the formula 

 may also be written, 



cos h = sin t. sin p. cosec Z (2) 



or in logarithmic form as follows: 



log cos h = log sin t + log sin p -\- log cosec Z (3) 



So that the logarithmic cosine of the required altitude is found by 

 adding together three logarithms, and all the logarithms used 

 are taken from one table, which is No. 44 in the collection of 

 mathematical tables of the A^nerican Practical Navigator issued 

 by the Navy Department at Washington. But, if an occasion 

 should arise in which logarithmic tables were lacking, it would 

 not be laborious to solve this problem by means of natural func- 

 tions alone as indicated in the equations numbered 1 and 2. 



The difference between the altitude, thus calculated as the al- 

 titude that the observed body would have if the observer stood in 

 the assumed geographical position, and the corrected altitude 

 measured in the actual position of the observer, gives the length 

 of the intercept, or altitude-difference, which, when laid off from 

 the assumed position, along the direction of the azimuth and 

 toward or away from the direction of the observed body ac- 

 cording as the observed altitude is higher or lower than the cal- 

 culated altitude, marks the point through which the line of po- 

 sition of the observer passes at right angles to the azimuth of the 

 observed celestial body. The first differential of h with respect 

 to Z, obtained by differentiating equation (2), gives the following 

 relation: 



— = cot h cot Z 



dZ 



