THEORY OF POLYCONIC PROJECTIONS. 159 



which is obtained by equating two expressions for the 

 ratio of the areas of the triangles. The same relation fol- 

 lows at once analytically from the tangent relation first 



given. The angle u increasing from zero to ^, its alteration 



u — u' increases from zero up to a certain value co, then 

 docreases to zero. The maximum is produced at the 



moment when the sum u + u^ becomes equal to k* -^^^ ^ 



and Z7' be the corresponding values of u and u\ We find 

 from the tangent formula that the following are their 

 values : 



tan U=n^ and tan U' = ~j=' 



-y/d VC 



The quantity w can be computed by any one of the f ormidas 

 c — d 



sm co = 



cos 03 = 



c + d 



24cd 

 c + d 



c— d 



, CO -J c — ^ 



tan(|+|) = ^ and tan(|-|)=^. 



From the last two equations since the sum of V and V is 

 equal to ^ and their difference .is equal to co, we have 



Z7=|+|. and V' = \-\. 



From the tangent relation we see that when we change u 



to o— 1^' it is sufficient to change u' to o"'^- ^^^ same 

 substitutions being effected in u-^-u' ^ give for result 

 TT— (u+uO, so that the sine formula shows that the value 



