THEORY OF POLYGON IC PROJECTIONS. 125 



If ip and ip' are to become -y simultaneously, we shall 

 have the condition 



and in this case the pole will be represented by a single 

 point. The equation then reduces to 



If to this equation we add the following: 



X = ^ f 3 + 1 an^ 2" ) *^^ "9 » 



we know that the surfaces will also be preserved along the 

 Equator; this equation was derived from the differential 

 equation 



^ = 3^(l+cosX)^ 

 which gives n"=^ when in it we make X = 2j ^'""|' ^^^ 



This value of n" gives 



(ZX=^ 



jK = i^3ir. 



The values for the magnification along the meridians and 

 parallels now become 



j^^= V3^ 1 



2 1 + cos <p cos X' 



2 (1+cosXO^ 



P V^TT 1 + COS <p COS X' 



and from these we derive 



^^/ l+cosV Y. 



VI + COS tp COS X / 



