THEORY OF POLYCONIC PROJECTIONS. 165 



The ratio of scale is given, by the equations 



or 



r cos u' =d cos u. 



If it is desired to determine the azimuth of the line from 

 a point to a near point from their coordinates on the 

 map, we have approximately 



x' 



x' and y' being the coordinates of one of the points with 

 respect to the other as origin in the transformed system; 

 that is, after the axes have been turned to make the axes 

 of the ellipse coincide with the axes of coordinates. Then 



tan u = — tan u^' , 

 c 



The azimuth reckoned from east to north is given by 



If the map does not extend more than 5 degrees beyond 

 the central meridian, the angle r] can be considered zero 

 and the reductions become comparatively simple. 



The theory of the elUptic indicatrix can be applied to 

 any projection that has a change of scale at any point 

 for different directions; that is, for any projection that is 

 not conformal. It has been applied only to the ordinary 

 polyconic projection in this publication, since for practical 

 purposes that one is probably the most important of the 

 nonconformal projections treated under the polyconic pro- 

 jections. 



The appended tables of the elements of the ordinary 

 polyconic projection are taken from Tissot's work. They 

 are computed for the sphere but can safely be used for 

 ordinary computation work. If more exact results are 

 desired the computations should be made from the first 

 by employment of the spheroidal formulas. 



