THEORY OF POLYCONIC PROJECTIONS. 121 



The area of the lune upon the unit sphere is equal to 2X; 

 hence by equating this area to the area of the projection 

 of the same we obtain 



X= . ^^, — cot X'. 

 sm^X' 



By computing by means of the first equation the values of 

 (p, which correspond to a sufficient number of values of <^', 

 we could construct a table which, reciprocally, would make 

 known the values of <p' corresponding to given values of <p. 

 The second equation would make it possible to solve the 

 same problem with respect to X and X'. 

 With these relations we obtain 



dip' TT COS <p(l — COS 2(^0^ 



d(p 4 sin 2<p' (2^' — sin 2^') 



dy ^ sin^X^ 



d\ 2(l-X'cotX0 



T _ TT COS (p sin (p' tan (^' sin B 

 V2 sin X' {2ip' - sin 2<p') 



, _ 1 sin X^ sin d 



P~^ cos <p tan <p' (1 —X' cot XO 



or 





TT COS ip tan ip' 



2 V2 o / '^ • 1+ COS X' cos <p' 

 ^ -V ^ 2<^' - 2 sm <^ 



7, _Jl_ cQS ip' sin^X^ 1 



^~ V2 ^^s <^ ^~^' ^^^ ^' 1 + C0S X' cos <p' 



_ 1 cos <p' 1 1 ^ 



~~^ COS V? 1-X cot X' 1+cos X' cos <^'' 



