THEORY or POLYCONIC PROJECTIONS. 27 



This equation shows that the parallels are circles, and that 

 the parallel of latitude ip has the radius a cot (p, and that 

 the center lies at the point a; = 0, y = a cosec ip. The paral- 

 lels are therefore circles, nonconcentric, but having their 

 centers on the hne ic = 0. The projection is thus seen to 

 be a polyconic projection in the sense of Tissot's definition. 

 By solving the original equations for sin <^ and cos ip we 

 find 



y sin X 



sm <p= 

 cos <p~ 



a sin X — a; cos X 



X 



a sin \ — x cos X 



By squaring and adding, the equation of the meridians is 

 obtained. 



y^ sin^X ^2 ^ 



(a sin \ — x cos X)^ {a sin X — a; cos X)^ ~~ ' 



or, on reduction, 



3-2 _^ ^2 _j_ 2aic cot X = a^ 



or, as usually written, 



(ic + a cot X)2 + ;z/^ = a2 cosec^X. 



The meridians are thus seen to be circles also ; the circle for 

 the longitude X has the radius a cosec X, and the center Hes 

 at the point x = a cot X, y = 0. 



In this projection we have, therefore, 



p = a cot ip 

 s = a cosec (p 



X sin X sin <p 



sm e 



p 1 + cos X cos <p 



de_ sinX 



dip 1 + cos X cos (p 



-T— = —a cot <p cosec cp 



dd ds . ^ a sin X cot (p a sm X cot (p 

 o<P dip 1 + cos X cos (p 1 + cos X cos (p 



