THEORY OF POLYCONIC PROJECTIONS. 89 



But on the earth 



(' 



dS,y_ a? cos" <p 



from which it follows that 



._dS_i 2A^ Vl-e^sin^ (P 



^~ dS~ a cos ip iA'e'^'' + 2AB cos 2l3\ + B'e--'^'^y 



In order to derive the equations in their usual form, 



we shall move the origin down to the point — ^d* The value 



of X will remain the same, but the new value of y will 



equal the old value of y increased by ^d or V' = ^+oo* 



The equations are. therefore, 



A sin 2/3X 



^ = ^2g20<r + 2AB cos 2i8X + B^e-^^'' 



_ J[2g2ffa_^2g-2^a 



^~2B {A^e^^'^ + 2AB cos 2/3X + 5^6-2^'')* 

 The equation of the meridians now becomes 

 / cot2^XY ,^ 1 



and that of the parallels 



2 r ^2g4ff. ^^2 -12 ^2g4^. 



^ "^L^ 2B{A'e^^''-B')j ~{A'e^^'^-By' . 



To identify this projection with the one formerly 

 obtained, let 



1 A 



— =c, 2^=n, and ^ = ^' 



Then 



2clc sin n\ 



■ p^<T _j_ 2]c cos n\ + e-""" 



y - 12 ^na j^ 21c COS n\ + e-""" 

 (x + c cot nXy + y^ = c^ cosec^ n\ 



^ "^L^ P62na_l J -(pg2n._l)2 



