THEORY OF POLYCONIC PROJECTIONS. 



By substituting this value, we obtain 



tan rj/ = sjl — e^ tan (p, 



sin ^=s 

 COS ^ = 



1 + tan^i^ — e^ tan^i^ 1 — e^ sin^,^ 



If we denote the radius of curvature PK of the meridian 

 by Pm, we have from the general theory of plane curves 

 the relation pjodip = ds. 



But 



ds = -^dx^ + dy^ = -yja^ sin^^ + h^ cos^^ drf/ = a-^l — e^ cosV <^^. 



Also « 



vr 



and 



vi-''cos^^=vr^.3.;„.. 



e' sm^(^ 



or 



Hence 



V ^ ^ (1— e^smV)'' 



^"■(l-e^sinV)''"* 

 a(l-6^) 



The normals at any two points on the same parallel circle 

 intersect in a point K^ of the axis of rotation. If we pass 

 a plane through these two normals and then let the nor- 

 mals approach each other imtil they finally coincide, we 

 obtain a vertical plane tangent to the given parallel and 

 perpendicular to the meridian at the point of tangency. 

 The radius of curvature of a small arc in this direction is 

 given by PK^ because the normals of two contiguous 



