XXI 



Then, if the directions of x be longitudinal from stern to head, y trans- 

 verse to starboard, z vertically downwards, we have 



X = Hcos, Y=-Hsin, Z = Htan0, 

 X' = H' cos ', Y' = - H' sin '. 



Resolving along and perpendicular to the direction of H we find, after 

 some reductions, 



. . (3) 



where 



Dividing the first by the second, of (3) we find 



which gives the deviation on any given magnetic course, , when the 

 five coefficients &, 23, C, !?, < are known. Multiplying both numbers 

 by the denominator of the second member, and by cos 3, and reducing, 

 we find 



sn = cos sm cos sm cos 



or 

 sina=^cosa-fBsmr + Ccosr + iism(2r + ^ + ^cos(2r + a). . (7) 



These give the deviations expressed nearly, though not wholly, in terms 

 of the compass courses. 



When the deviations are of moderate amount, say not exceeding 20, 

 equation (6) or (7) may be put under the comparatively simple and con- 

 venient form 



s2', .... (8) 



in which the deviation is expressed wholly in terms of the compass 

 courses ; and this will be sufficiently exact for practical purposes. 



It will be seen that the 3P, JJ, C, 3B, $ are nearly the natural sines of 

 the angles A, B, C, D, E. 



