THEORY OF POLYCONIC PROJECTIONS. 39 



By solving the original equations for sin X and cos X we 

 get 



x(sin a + sin ip) 



sm X = 



a sm a cos (p + y cos a cos ip 



a cos a sm (p — y — y sm a sm <p 

 ~ a sin a cos <p-\-y cos a cos ^ 



By squaring and adding we obtain 



a;^(sin a + sin <py+(a cos a sin (p — y — y sin a sin <^)^ = 

 cos^ ^(a sin a + y cos a)^, 



or, on developing and arranging, 



a^^Csin « + sin <^)^ + ^^(sin a + sin (pY — 2ay cos a (sin a + sin (^s) 

 = a2(sin^ a cos^ ^ — cos^ a sin^ <p) 



or, finally, 



/ g cos a Y _ g^ cos^ <p 



^■^ V ~sin a + sin <p/ ~ (sin a + sin <py' 



The parallels are, therefore, circles with their centers all 

 lying on the Y axis. The parallel of latitude (p has the 

 radius 



a cos <p 

 ^^ sm a + sm ^ 



with its center at the point 



x = 0, 



a cos a 



y= 



sm a + sm (p 



The parallel of latitude —a is evidentl;^ a straight line, 

 since the radius becomes infinite for tms value, as does 

 also the distance of the center from the center of the 

 projection. 



The projection is seen to be a polyconic projection in 

 accordance with the definition of Tissot. 



