80 U. S. COAST AND GEODETIC SURVEY. 



Since this equation contains only X and is independent of 

 (p and ^, it is the equation of the meridians. The meridians 

 are therefore circles with centers upon the X axis (the 

 straight line parallel of the map) lying at the distance 

 = —G cot tCk from the origin and having the radius 

 = c cosec nX. 



Since f or £c = 0, y = ± c, all of the meridians pass through 

 the two points which are distant +c and —c from the 

 origin; 2c is therefore the length of the central meridian 

 included between the poles. 



As an aid to construction, we may assume the equation 



itan^|+f) = tan(^+|); 



then 



8 = e cosec ^ 

 and 



p=^c cot ^. 



A special case of this projection is given by the values 

 A: = 1 and n = 1 • in which case ^ = (^, and 



s = c cosec ip 



p=^c cot (p 



and the equation of the meridians becomes 



y2 + (a; + c cot X)2 = c^ cosec^ X. 



This is evidently the stereographic meridian projection, 

 which has already been discussed under that heading. 



DETERMINATION OF THE CONFORMAL PROTECTION IN 

 WHICH THE MERIDIANS AND PARALLELS ARE REPRE- 

 SENTED BY CIRCULAR ARCS. 



This projection is the one devised by Lagrange. His 

 problem was to determine the general conformal projec- 

 tion in which the meridians and parallels were both 

 represented by circular arcs. 



Since the projection is to be conformal, we can express it 

 in the form of a function of a complex variable.* 



♦See The General Theory of the Lambert Conformal Conic Projection, Special Publica- 

 tion No. 53, U. S. Coast and Geodetic Survey. 



