34 U. S. COAST AND GEODETIC SURVEY. 



We can now^map the ellipsoid conformally upon the 

 sphere by the relations 



X' = X 

 and 



*-(-M')=*-(M) • (feS)^- 



The latitudes ip' are computed for the parallels that we 

 may wish to map; that is, for 10°, 20°, etc., or for what- 

 ever interval we may choose. This sphere may then be 

 conformally mapped upon the plane, the values of ip' being 

 employed in the computation. Each step is conformal; 

 hence the plane map is. a conformal representation of the 

 ellipsoid. 



The magnification upon the sphere is given by 



dZ a cos ip 



-'(^S^-*-)' 



a cos 



(1 - e sinV)^^ LcosV (1 - e' sinV)' "^ J 



_ cos (p' (1 — €^ sin^v?)^ 

 cos v? 



The total magnification is equal to the product of the 

 values obtained for the ellipsoid upon the sphere and for 

 the sphere upon the plane. The total magnification, 

 which we shall denote by Ic without subscript, since it is 

 the same at any point in all directions, is given in the form 



,^ cos V?' (l-c^sinV)^ 



cos (p (1+cos X cos ip') 



CONSTRUCTION OF STEREOGRAPHIC MERIDUN PROJECTION. 



It is a very easy matter to construct a stereographic 

 meridian projection graphically. Divide the meridian 

 circle into equal arcs at whatever interval it is desired 

 to construct the meridians and parallels. In figure 8 the 

 divisions are made at 30° intervals. Q,E' = 30° ; the tangent 

 at R' gives the radius 8'R' and the center S' iov the 

 parallel of 30°; a similar arc with center distance to the 

 south e^ual to OB' and with radius equal to S'W gives 

 the projection of the parallel of 30° S. The tangent at 

 R or SR gives the radius for 60° of latitude, and the 

 same arc transferred to the south gives the projection 



