26 U. S. COAST AND GEODETIC SURVEY. 



P is the point that we were considering in the previous 

 fisrure. 



PQ=V 



APTQ = \ 



From the trigonometry of the spherical triangle we 

 have the relations 



cos p = cos X cos <p 



sin p sin co = sin (p 



sin f cos CO = sin X cos (p. 



If these values are substituted in the equations for x 

 and y, we obtain 



a sin X cos <p 



X = z: 



y 



1 + cos X cos <p 



a sin <p 

 l+cos X cos cp' 



From these equations, by solving for sin X and cos X, 

 there result 



sin X=- tan <p 



Hence 



^ a sm (p—y 



GOS X = —^ -' 



y cos <p 

 y^ ^ y^ Q.os^(p ' 



or, by reduction, 



3.2 ^ y2 __ 2ay cosec (p=^a? 

 or, as usually written, 



x^+(y — a cosec <py = a^cot^^. 



