THEORY OF POLYCONIC PROJECTIONS. 95 



If n becomes equal to unity, we obtain the stereograpbic 

 projection and the equations take the form 



2cm sin X 



1 + 2m cos X + m^ 



c(m2-l) 



^ l + 2mcosX+m^ 



with m = tanf 5 + 2) ^^^(4 + 2) 

 Substituting this value of m and reducing, we obtain 

 c cos a sin X cos (p 



x = 



1 + sin a sin <p + cos a cos X cos <p 



c (sin a + sin (p) 



^ "~ 1 +sin a sin (p + cos a cos X cos (p 



If we now let y' = y — sin a, which merely moves the origin 

 and does not change the nature of the projection, we 

 obtain after dropping the primes 



c cos a sin X cos (p 

 x= 



1 + sin a sin (p + cos a cos X cos (p 



c cos q:(cos a cos <p — sin a cos X cos <p) _ 

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



Now by replacing c cos a by a, we arrive at the values pre- 

 viously obtained 



a sin X cos <p 

 x= 



1 +sin a sin <p + cos a cos X cos (p 



_a(cos a cos cp — si n o; cos X cos (p) 

 ^"~l+sin a sin ^ + cos a cos X cos <p 



