Grinten — Projection of the Whole Earth) s Surface. 359 



lels are to run as straight lines parallel with the equator, we 

 have y = d, in which case, although equivalence and conform- 

 ity are preserved along the equator, © increases to 90° at the 

 poles (?? = 0) ; and finally if it is required that the deformation 

 (k) of any parallel at the marginal meridian shall be equal to 

 its deformation at 2 



the central me- 

 ridian, in which case 

 all the parallels are 

 practically inter- 

 sected by the me- 

 ridians at equal dis- 

 tances, we get 



7 3-<f 

 y =z a — - — . 



J 2 



By the first arrange- 

 m e n t (rectangular 

 trajectories) the 

 parallels at the mar- 

 gin would approach 



too near to the poles 90 ° 



(h representing the geometrical construction 



deformation of me- Pa+enj-ed 'in U.S. Canac/a^Gr.Britain^rance^ 

 ridional parts, <f> the 

 latitude, and X the longitude) : 



h 



= 



A = 180 



2; h 



= 90 



2 = 180 



= 90 

 A = 



in the second case they would appear to be too far distant 

 from the poles : 



^ = 90 

 2 = 180 



.h 



= 90 

 2 = 



finally in the third case we are confronted with an unhappy 

 congestion of the temperate zones : 



= = 90 . * = 90 



h = f ; n = a/ 3* h 



2 = 180 2 2 = 180 V 2 = 



Therefore it seems desirable to find a middle ground by hav- 

 ing all alterations continuously increased from zero at the 

 equator to their least maxima at the poles : 



= 



180 



1; h 



90 



= 90 



iso - ^ 2 ' A 2 = o 



This arrangement has the effect of making the continental 

 regions fall within the zone of least " total error," though this 



