THEORY OF POLYCONIC PROJECTIONS. 43 



By substituting these values we obtain 



ai (tan <p — i sin X cosh /3 — cos X sinh jS) 



_ a sin X cos h ^ + ai (tan <p— cos X s inh jQ) 

 " sec v? cosh jS + tan (p sinh jS + cos X 



By equating the real parts and the imaginary parts, we get 



a sin X cosh B 



X = ^ 



sec (p cosh ^ + tan (p sinh /3 + cos X 



_ a (tan (p — cos X s inh /3) 



^~sec (p cosh jS +tan (p sinh jS + cos X 

 Let 



cosh /3 = sec a, 

 then 



sinh i(3 = tan a. 



Substituting these values we obtain 



a sec a sin X 



a; = 



^ 



sec a sec <^ + tan a tan (^ + cos X 



a (tan <^ — tan a cos X) 

 sec a sec (p + tan a tan (p + cos X 



On multiplying both numerator and denominator by cos oc 

 cos (p, we derive 



a sin X cos <p 

 x = 



1 + sm a sm <^ + cos a cos X cos <p 



_ a(cos q: sin <p — s in a cos X co s y?) 

 ^ 1 + sin a sin <^ + cos a cos X cos <^ 



We thus arrive at the same equations that were ob- 

 tained before. 



PROOF THAT CIRCLES PROJECT INTO CIRCLES IN STEREO- 

 GRAPHIC PROJECTIONS. 



It can be proved in a general way that, in any stereo- 

 graphic projection, any circle upon the sphere is projected 

 into a circle upon the plane of the map. Straight lines 



