32 TJ. S. COAST AND GEODETIC SURVEY. 



Thi& can also be written in the form 

 x + iy = ai tanh ( — ^ — ) 



ai sinh ( —^ — ) 



x + ^y = 



cosh 



m 



ai sinh l^-^ — ) cosh (—^ — ) 

 cosh {-^) cosh [-^) 



ai (sinh o- — sinh i\) 

 "" cosh (T + cosh i\ 



_ ai (sinh o- — '^ sin X) 

 ~ cosh cr + cos X 



_ a sin \ + ai sinh a 

 "" cosh <T + cos X 



a sin \ + ai tan <p 

 ~ sec (p + cos X 



_ a sin X cos <p + a^ sin <p 



"~ 1 + cos X cos (p 



By equating the real parts and the imaginary parts this 



becomes 



a sin X cos <p 



X == — 



1 + cos X cos (p 



a sin ip 



y = o 



^ 1 + cos X COS (p 



We thus by this method arrive at the same values that 

 were obtained before by expressing analytically the results 

 of the direct projection. The fact that the projection can 

 be derived by the use of functions of a complex variable 

 establishes the conformality of the projection * 



♦See Coast and Geodetic Survey »Special Publication No. 53, The General Theory of the 

 Lambert Conformal Conic Projection. 



