164 U. S. COAST AND GEODETIC SURVEY. 



If 7 is the angle between the conjugate axes, and if 17 

 is the angle between the major axis and the conjugate 

 axis of X. we have from the theory of conjugate axes 



d^ 



tan 17 tan (77 + t) = — -2 • 

 c 



By developing this expression we get 



^ dJ^ + c^ tan^ 77 



tanT=-(,._^.)t^^^; 



but 



7=1 + ^. 



Therefore 



, , (Z^ + c^ tan^ r; 

 cot^ = (c^_<i.)tan,- 



By solving this for tan 77 we get 



c^ — dj^ (c^ — d^y d^ 



from which r; can be determined. The angle between the 

 minor axis and the conjugate minor axis is equal to 17 + ^. 

 If ^ is counted positive for points east of the central 

 meridian, the axes must be turned through the angle 

 ^ — r}~\l/. We shall then have 



x' =x cos (^ — >7 — tA)+y sin ($—17—^) 



y'=-a;sin (^-77-1^)+^ cos (^-i?-^). 



For points west of the central meridian ^— ?? — ^ can be 

 considered negative in the transformation formulas. 



If geodetic azimuths are ^iven, they should first be 

 referred to the parallel as initial line; that is, they should 

 be reckoned irom the east around counterclockwise 

 through north. If the rj + xj/ angle is added to these 

 azimuths we shall obtain the an^Ie u. Since the elliptic 

 indicatrix has the minor axis m the direction of the 

 initial line, we have 



/» 

 tan t^=-j tanifc. 



