THEORY OF POLYCONIC PROJECTIONS. 15 



Since for X = 0, ^ must also be zero, the function r(X) must 

 vanish with X. This is the only condition that is required 

 to give a rectangular poly conic projection. 



If we choose an arbitrary function for r(X) that van- 

 ishes with X and another arbitrary function of ip for u and 

 set 



* , ^ r(x) 



tan x-= 1 



2 u 



then the net will always be rectangular provided that 



ds 



"^^Ai' 



d(p 

 du 



dip 



in which s is also an arbitrary function of (p, or provided 

 that 



=/ 



p du 7 

 u d(p ^ 



with p arbitrary. 



Since in this case of the rectangular polyconic projec- 

 tion ^ = and sec 1^= 1, we have 



_ (1 - e2 sin2 cpyi^ {ds ^„„ ^ dp 



Km — ■ 



a{l-^) 



/ ds /^_^^ 



\d(p dip) 



since 



P a cos <^ r(X) ' 



^^ r'(x) . ^ 



- sm Q, 



d\ r.(x) 



If we wish the parallel of latitude (p to lie on the developed 

 base of the cone tangent to the earth at latitude cp, we 

 must have 



a cot <p 



{l-e^siit^ (pyi-^ 



If, besides, the parallels are to be spaced along the central 

 meridian in proportion to their true distances, we must 

 also take 



"Jo (1- 



(1 — e^) dip a cot (p 



t^sm" ipY'^ ^ il-e'^sin^ <pyi^ 



