THEORY OF POLYCONIC PROJECTIONS. 109 



DISCUSSION OF THE MAGNIFICATION ON THE CONFORMAL 

 DOUBLE CIRCULAR PROJECTION. 



The values which we have found for Icm. and ^p in any 

 system of rectangular projections with circular meridians 

 and parallels have now become equal to each other and 

 we have for the ratio of the lengths at each point of a 

 conformal projection 



7, _ '^ si^ ^ 



cos (p tan (p' sin X' 



It results from this ec[uation that, upon any ^iven parallel, 

 Ic increases or diminishes at the same time as X. When 

 the value of sin d is substituted, we obtain 



7„ _ '^ ^^^ ^ 71 sin y' 



sec <^' + COS X' sin 'p (1 +cos X' sin p') 



A point of discontinuity is found when cos X' sin v' = —1^ 

 Within the limits of the map this can happen only when 



TT ... 



f' =2 and X'=±7r. In the stereographic projection this 



point is the antipode of the center of the map. If ti is 

 less than unity it would fall outside of the map of the 

 whole surface; but if n is greater than unity it would fall 

 inside of the map of. the earth's surface, since we should 

 have nX = ± TT. 



For convenience we will write the above expression in 

 the form 



n 



in 2> Kl ^^^ 2 "^ ^^^ 2 /"^^^^ ^' ' 



In this expression we need only to replace X' by n\ and 



tan 9- by ( cot ^ tan f ) to obtain Tc directly as a fimction 



of 2> and X. In order to see immediately what happens to 

 Ic at the poles, we shall make this substitution and express 

 the result in the form 



i-(«.'f);(-'-i)'"(""ir 



I tan ^ ) ( sin ^ ) ( cos ^ j +sin f cos n\ 



