Marine Collima ting-Compass 179 



Let the following notation be adopted : 



r = the mean of a number of scale readings.' 

 h = the altitude of the star. 

 V = the value of one scale division, in degrees. 



VI = the altitude of the scale, positive above the horizon, negative below, 

 the angle SZS' = NZN' = D, the magnetic declination. 

 " " SZs = A^, the astronomic azimuth of the star. 

 " " S'Ziv = Ac, constant for the scale in question. 

 " " wZw' = r — 5, the mean of a number of scale readings, less 5. 

 the arc w's = A, the arc measiu-ed by the sextant, corresponding to r. 



" " sZ = 90° — h, the apparent zenith distance of the star. 



" " ivZ = w'Z = 90° — m, the apparent zenith distance of the scale, 



the angle tc'Zs = A, the horizontal angle between scale and star. 



The magnetic declination may then be expressed by the equation 



D =^ A, - [Ac + {r - 5.00)y ± A] (1) 



The terms Ac and v of this equation are constant for each scale of the instrument. The 

 term A is computed from the spherical triangle iv'Zs, of which the side w's is directly 

 measured, and the sides w'Z and sZ are known from the elevation or depression, m, of the 

 scale, and the apparent altitude, h, of the star. The constants m, Ac, and v are determined 

 at magnetic comparison-stations. The apparent altitude, h, may be observed directly or 

 it may be computed. When obtained by observation at sea, it is to be freed from dip of 

 the horizon, but if the true altitude is computed, it is to be increased by the corresponding 

 refraction correction. 



The angle A, or the azimuthal difference between the scale and star, has been given a 

 double sign in equation (1) since the star maybe to the right (-|-) or left ( — ) of the scale. 

 It may be computed from any one of the following fundamental formulae of spherical 

 trigonometry, in which m may be either positive or negative : 



cos kA = Vcos (s — A) coss sech seem 



A = V sin (s — m) sin (s — h) sech seci 



sin X A = V sm (s — m) sin (s — ti) secft seem 



tan n -4 = V sin (s — m) sin (s — h) sec (s — A) sec s 



. _ cos A — sin/i sinm ,„« 



cos A — \Ji) 



cos h cos m 

 In the above equations, 2s = A + m + h. If m = 0, they reduce to 



cos ■^A=^ cos ^(h — A) cos ^{h + A) sec h 

 sin X .A = ^ sin ^ (A — /i) sin ^ (A -|- h) sec h 

 tan ^ A = -Jtanx (A — h) tan^ (A + 



h) 



cos A = cos A sec h (3) 



Equation (3) is convenient for logarithmic computation, and though the angle is given by 

 its cosine, it is sufficiently accurate for five-place logarithms, when A is over 30°, and the 



'The scale diviBions are mentally numbered from left to right, the middle division being 5. 



