48 U. S. COAST AND GEODETIC SURVEY. 



and for the parallels 



a cos (p a cos ^ 



''P sin a + sin 



a cos a a cos a 



^^~~sin a + sin 



- 2sin(^)cos(^) 



- 2 sin (^-) cos (^-). 



The forms last given should be used for logarithmic com- 

 putation. 



CONSTRUCTION OF STEREOGRAPHIC HORIZON PROJECTION. 



The method of graphical construction for the parallels 

 is as follows: Let us suppose that we wish to construct a 

 projection for a =30°. In figure 12 the point of projection 

 is supposed to be in the perpendicular to the plane of the 

 paper at E. Let the plane of the central meridian (that 

 through the point of projection) cut the mapping plane or 

 the plane of the paper in the line YY' . This central 

 meridian section is then turned upon YY' as an axis until 

 it falls in the plane of the paper. The eye will then be at 

 0, and A will be the point that projects into the center of 

 the map. Construct the angle AEQ equal to 30°; then 

 QQ' is the trace of the equitorial plane upon the plane of 

 the central meridian. The diameter PP' perpendicular to 

 QQ' is the axis of the eaipth turned with the plane of the 

 central meridian. YY' is the projection of the central 

 meridian, since the plane was turned upon this liQe as an 

 axis; hence, if any point is projected upon this line the 

 corresponding point upon the map will be determined. 

 P and P' are the poles; draw OP and OP'. Then jp is the 

 North Pole of the map and jp' is the South Pole of the 

 same. 



To determine the circle that forms the projection of any 

 parallel, lay off the arc CQ equal to the latitude; in the 

 figure 0^=45°. Construct CB perpendicular to PP' and 

 construct tangents at B and C meeting in the axis pro- 

 duced at D. Draw OB, OC, and OD] then B' and c' are 

 points on the circle, and D' is the center of the same. 

 With D' as center and with radius D'B' or D'c' construct 

 the circle, and the circle so drawn in the figure is the 

 projection of the parallel of 45° of latitude. OQ deter- 



