104 TJ. S. COAST AND GEODETIC SURVEY. 



A\ B', C, and D' being constants depending upon the 

 position of the center and the radius of the circle. In 

 the meridian e+'^reographic projection we have <p' = <^ and 

 X' = X, so that it is only necessary to take A', B' , C , and 

 D' proportional to A, B, C, and D, respectively, in order 

 that the two circles may correspond to each other. ^ There- 

 fore, in the stereographic projection on a meridian, and 

 as a consequence also upon the horizon of any place, 

 every circle is projected into a circle. This fact has 

 already been proved in another place by the use of ana- 

 lytic geometry.* 



Let us now determine the expressions for the scale 

 along the meridian and for that along the parallels. When 

 the point Jf is displaced infinitesimally upon the projection 



of the meridian, the arc described is equal to ^(^—7) d^' ■ 



and when displaced upon the parallel the arc described is 



equal to n ^/) d^'\ therefore, we have 



-<S) 



d(p 



tCr, 



r fdl\d>/^ 



'P cos <^\dXv d\ 

 Now, if we take the logarithms of the two members 

 of the formula which gives tne value of tan ^ and then 

 differentiate, we obtain 



dd dy , dip' 



y+-:A. 



sin d sin X' sin ip' 



which gives for the partial derivative values the following 

 expressions : 



be ^ sin e , be ^ sme 



d<^'~sin^' dX' sin X' 



On substituting these values and the values of r and R 

 we obtain 



, sin e dap' 



"""sinX' sin<^' d<p 



sin 5 dk^ 



P ~ cos v? tan (p' sin X' (ZX ' 



*Seep. 43. 



