THEORY OF POLYCONIC PROJECTIONS. 107 



stereographic meridian projection. The parallels are 

 determined by the equation 



tan Tr = 



, tan-§ 



2 tan I 



Parallels constructed for p' on the meridian projection are 

 the parallels for jp on the horizon projection. The circle 

 constructed with its diameter consisting of the chord for 



<Pq = 'Po — 2 ^^ ttie meridian projection becomes the projec- 

 tion of the horizon circle in the horizon projection. ^ In 

 figure 32, pMp'N is the meridian circle of the original 

 meridian projection and PQP'Q' is the horizon circle for 



27r 

 Po=~S^ constructed on the chord of the meridian circle for 

 o 



(Pq=-f' Tangents to the computed p' points of the meridian 



circle would determine the centers and radii of the arcs 

 for the horizon projection; or the radii and center dis- 

 tances can be computed from the expressions for r and s in 



terms of <^' = -2— 2>'' 



If we let ^0 become -^ and then let n converge to zero 



while leaving constant the product of n hj the length OP in 

 figure 31, which we have chosen as unity in the former 

 analysis, we obtain again Mercator's projection. If we 

 maintain this product equal to two, we shall have con- 

 stantly 



tan-s- o 1 



06^ = X-rT^»andOZ>=- - 

 A n 



-(tanf)° 

 l+(tanf)° 



The limiting values of these expressions as 7i=0 are given 

 in the form 



0G=^\ and Oi> = loge cot |.* 



* For the derivation of these limits see p. 94. 



