THEORY OF POLYCONIC PROJECTIONS. 17 



This value gives 



, e X . 



tan = 9 sin <p. 



This gives the full determination of the projection. With 

 these values we shall now determine the magnification 

 along the meridians and parallels. 



r'(x)=i 



dp a cosec^ <p a^ cos^ ^ 



"3^" ~ (l-e^sinV)'^^ (1 - e^ sin2 ^^h 



_ — g cosec^ ip-\-ae^^a^ cos'^ (p 

 ~ (l-e^sin^iiP)'/^ 



and 



ds _ a Q,o\? (p 



d<p~ iX-e^^m^ <pyi^' 



Substituting these values in the differential formulas on 

 pages 12 and 13, we obtain 



-, cosec^ v? €2(l + cos2v') 1— e^sin^^ 



km= j^_^2 f3"^i iZr^i COt^ V? COS e 



, _ sin ^ 



^ X sin <p 



The formula for ^m shows that the value of Z:^ along the 

 central meridian is equal to unity; that is, the sc3e is 

 maintained constant along this meridian as was provided 

 by the choice of the value for s. This means that the 

 parallels are spaced along the central meridian in pro- 

 portion to their distances apart upon the earth. Since 

 this is true, with the known radii we can construct the 

 parallel arcs either by drafting or by plotting by means of 

 computed coordinates. The only things remaining to be 

 determined are the points of intersection of the meridians 

 with these parallels. 



In order to determine these points, we have first 



^ _ a\ cos (f 

 ^ ^^^2~2(l-€2sin2^)V2' 



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