THEORT OF POLYCOXIC PROJECTIOXS. 105 



or, on substituting the value of sin 6, 



J, = 1 ^' 



'™ 1+cos X' cos <^' cZv? 



J 1 cos (p' dy 



^~ l+cosy cos <p' cos (p d\' 



CONFORMAL DOUBLE CIRCULAR PROJECTIONS. 



In the conformal polyconic projection the condition 

 ^m = ^p gives in the case of the double circular ortho- 

 gonal net 



sec <p^ d(/_dy 

 sec <p dip d\ 



The left-hand member of this equation is a function of 

 (p alone and the right-hand member a function of X alone; 

 it is therefore necessary that they should be equal to the 

 same constant n; hence 



dk' = n d\ 

 and 



dip' _ dip 



7 — 77/ 



COS (p cos <p 



By integrating the first equation we get 



y=n\, 



no constant of integration being introduced, since X' 

 vanishes with X. In the second equation let ip' = -^ — f' 



IT 



and let ip = ^ — p and we obtain 



sm p sm p 

 Let us write this in the form 



cot ^ -^+tan'^ 2 ^^ cot h -^ + ^ tan 22' 



