THEORY OF POLYCONIC PROJECTIONS. 127 



In the parentheses of the second member the factor which 

 varies with ^' is 



1+cos X^o cos if ' _ cos X^Q — cos X^ 

 1+cos X' cos ip' cos X' + seo ^'* 



We see, then, that upon each of the meridians for which we 

 have X<Xo, the ratio K'\^ less than unity and increases from 

 the Equator to the pole ; for X > Xq we have K> 1 and K 

 increases from the pole to the Equator. We should see in 

 a similar manner that, upon each parallel whose latitude is 

 less than <^o; -^ is smaller than unity and increases with the 

 longitude, while, if <p is greater than <^o> ^ will be greater 

 than unity and will increase as the longitude decreases. 

 Thus ^attains a minimum K^ at the center of the map, and 

 another K^ at the pole on the principal meridian; it attains 

 a maximum K^ at the pole on the central meridian; and, 

 finally, a second maximum K^ at the intersection of the 

 Equator with the principal meridian; these values are 



J. r (i+cosx^o) (1+cos^^on ^ 



^^^L 2(l+cosX'oCOSv?'o) J 



(1 +COS X'o cos ^'o)^ 

 ^( 1+COS <p'o Y 



^ Vl+COS X'o COS ip' J 



^( 1+cos x^o y 



* \1 +COS X'o COS «^'o/ 



Let US stni consider the rectangular circular projection 

 in which the hemisphere is represented by a complete 

 circle, and let us now suppose that we wish to develop 

 the central meridian with its true length. In order to 



do this we take the radius of the map equal to 2* Iii 



figure 30 we have seen that the three points A'^p^ and 

 JJ are in a straight line; hence the angle OA'D is equal 



to the half of <p\ Moreover, we have here OA' = -x and 



OD = <p] the right triangle OA'D will then give 



