THEORY OF POLYCONIC PROJECTIONS. 137 



The latter formula is very easily deduced, since by 

 logarithmic differentiation we obtain 



1 dy 1 



sin X' d\ X' 



when this value is substituted in the general formula, we 

 obtain the relation as given above. The formula for ^m is 

 somewhat more complicated in its derivation. We have 

 from the a priori conditions 



s — r = <p 

 or 



From the triangle OSU we obtain 



ir = sr-{-^ — ']rs sm <p; 

 but 



or 



When these values are substituted in the general formula on 

 page 134, we obtain the value of A:^, as given above. A 

 circle constructed upon the line of poles of the map as a 

 diameter gives the projection of the principal meridian. A 



