Grinten — Projection of the Whole EartWs Surface. 361 



mined, that y comes nearest to c = — for rectangular projec- 



tion, making ©, continuously increasing, a minimum. The 

 above expression then is 



7 a 2 + d<s/d i (\-d 2 ) + d i _ ,-— — -— — — 



y — d — a v ; and as V« (1— d 3 ) + d i 



Cv -f- w 



cannot exceed the unit (e? = 1 ; y = 1), y will become a maxi- 

 mum, under the limitation x $ = ° = oo , and, therefore, © a mini- 

 mum, just when V a 2 (1 — d 2 ) + d* is made a unit, giving 



a = \/l+d 2 , or x = ^ — , and finally y = 



d ' * 7 ^ 1— tf+ef 2— c 



d c 



and sin 6 = sin (yjr—2(j) / ) = - — - — -« = . Any other value 



1 -p 05 T~ 05 L. -f- C 



for a, less than VI +rZ 2 , would produce a maximum of © between 

 the equator and the poles. This maximum would reach its 



climax for a = 1, x = -j (® = at the equator and at the 



poles), the determination and location of which would require 

 the solution of a very lengthy equation. The requirement of 

 a minimal 2co involves a much more intricate equation still for 

 the determination and location of the maximal <& that is formed 

 between the equator and the poles. 



The harmonic relation between y, c and the radius of the 



2ty 

 marginal meridian (= unity) is then defined by c = . 



~ if 



Other harmonic relations occur at latitudes 72° and 54° as 

 °d 2cd 



y = T^ aud ^^i res P ective] y- 



I now offer the formulas of deformation in the most con- 

 densed form, which will furnish the necessary data for a table 

 showing the most characteristic features of the circular projec- 

 tion in a numerical way ; and which will enable the student of 

 cartography to extend the table to apply to any interval of 

 latitude whatsoever. 



Deformation. 



The deformation of an infinitely small part of the central 



meridian at a point ( — 1 is expressed by the ratio h = -=— , 



r representing the radius of any globe. It being necessary to 

 have h^ _ = k^ _ = 1 at the center of the map, we get 



