THEORY OF POLYCONIC PROJECTIONS. 133 



or 



. ,. , V sin 0- COS \J/ 

 sm(9 + .)^ cosX' 



.- , >. sin 7)' COS (7 cos \p 



cos {5 + (T) — — = -* 



sm 7 



It is, however, sufficient to calculate one of the angles d 

 apd 5; we have, in fact, 



for, 1 being the point of intersection of TZJwith PP\ the 

 two triangles 01 T and ISM have the angles at I equal, and, 

 by expressing that the sum of the other angles are the 

 same m the one triangle as in the other, we obtain the 

 relation which we have just written. 



The rectangular coordinates of the point M with respect 

 to the axes OA and OP are 



x = r sin 6 



y=R sin 8. 



^ sin p d\ 



By taking, with respect to p and with respect to X, the 

 derivatives of the logarithms of the two members of each 

 of the relations which we have established between the 



different variables, we obtain j- and vr , which figure in 



the values of A:^ and Jc^; but it is more simple to obtain 

 A^m by making use of the formula 



We now have 



/dr ds A , 



which has been demonstrated with regard to polyconic 

 projections in general. Since the meridians are also 

 circles with their centers upon the same straight line, 

 we can form an expression for Icp by replacing in the 



