THEORY OF POLYCONIC PROJECTIONS. 13 



The arc of a parallel on the map between the meridians 

 of longitude X and \ + d\ is equal to 



p ( >-r- j d\, since <p is constant. 



This arc upon the earth is equal to the expression 



J. a d\ cos <p 



Therefore 



7, _ P (l-e^sinV) V'»d(? 

 ^ a cos (p b\ 



The ratio of increase of area, denoted by Ky is given by 



K=JcJcp sin (Z-rpJ^lcmkp cos ^, 



or 



^_ p(l-e^sinV)^ /ds- ^^^ ^ _ dp\ dj^ 

 ^ a? (1 — e^) COS ip\di(> dip) bX 



CLASSIFICATION OF POLYCONIC PROJECTIONS. 



The general division of poly conic projections is sub- 

 divided into the following classes which are not, however, 

 mutually exclusive : 



(1) Rectangular polyconic projections. 



(2) Stereographic meridian and horizon projections. 



(3J Conformal polyconic projections. 



(4) Equal area or equivalent polyconic projections. 



(5) Conventional polyconic projections. 



(6) Ordinary, or American, polyconic projection. 



The general differential formulas developed above will 

 now be applied to these classes in tne order named. 



RECTANGULAR POLYCONIC PROJECTIONS. 



The condition that must be fulfilled if the meridians and 

 irallels of the map are 

 expressed analytically by 



parallels of the map are to intersect at right angles is 

 lITv' 



iA = 0. 



Since this condition requires, whatever the value of s and p, 

 that 



tan ^ = 0, 

 we must have 



p 5— -f J- sm ^ = 0. 

 o<p dip 



