THEORY OF POLYCONIC PROJECTIONS. 21 



By substituting these values we obtain 



r/ cosec^ A /^ , X^ . , \ eMl+cosMA ,V . , \ 



fZl^ — cot2 <pyl-^ sm2 ^jj-^^l+- sin2 ^j 



or, on reduction, 



1 I ^^ • 2 . ^^ 2 1— e^sinV 

 H-jsinV+2 cosV — ^_^2 



A = 



Lty^, we shall fi 

 curve along which there is no exaggeration of area. On 



If we equate this to unity^, we shall find the equation of a 



along which there is no exagrsreration of 

 reduction this equation becomes 



X^ sin^ <p + 4X2 sm^ <p - 8\' cos^ cp (^— j4r^) = ^^ 



which is satisfied by X = 0, or by the equation 



X2 sinV + 4 sinV-8 cos^ <p (^|^^^^) = 0. 



The areas of all sections north of this curve are diminished 

 and those lying south of it are increased in their represen- 

 tation on the map. 



If we confine ourselves to the consideration of the sphere 

 K may be expressed in the form 



1+J+-JCOSV 



(l + Jsinv) 



The differential element of area of the representation is 

 given in the form 



l+j + ;jCOsV 



dS = 0.2 -y r-2 r^ cos (p dip d\. 



(l+fsiavj 



