42 U. S. COAST AND GEODETIC SURVEY. 



The projection is thus shown to be conformal, since the 

 meridians and parallels are orthogonal and the magnifica- 

 tion along both is the same. We might have taken this 

 for granted since we found that the stereographic meridian 

 projection was conformal and the nature of the projec- 

 tion is not changed by moving the point of projection to 

 a different point upon the sphere. 



In taking account of the spheroid we proceed as in the 

 case of the stereographic meridian projection. The magni- 

 fication at a point (the same in all directions) would then be 



, cos ip' (1 — ^ ^-^ipY^ 



~cos ip (1 +sin ol' sin ip' +cos a' cos X cos ip') 



DERIVATION OF STEREOGRAPHIC HORIZON PROJECTION 

 BY FUNCTIONS OF A COMPLEX VARIABLE. 



The projection, being a conformal projection, can be ex- 

 pressed^in terms of a function of a complex variable either 

 of a+'iX or of 0- — iX. Let us take 



. . , /(7-iX-i8\ 

 a% smh ( o ) 



But 



a% sinh i - — ^ ) ^^^^ ( o ) 



ai [sinh (t — sinh {i\ + /3)] 

 cosh (o- + iS) + cosh %K 



ai [sinh a — sinh iX cosh ^ — cosh i\ sinh fl 

 cosh a cosh jS + sinh cr sinh jS + cosh ik 



cosh (7 = sec (p 



sinh (7- = tan (p 

 sinh i\=i sin X 

 cosh i\ — cos X. 



