THEORY OF POLYCONIC PROJECTIONS. 163 



which proves the statement. It results from this property 

 that the ratio of lengths in the two directions the angle 

 of which undergoes the maximum alteration is equal to 

 -^cd] for the angle which is not altered and which has for 

 side one of these two lines reduces to zero, and it has the 

 same line for second side, so that v' =r" = -yjcd. 



In the ordinary, or American, poly conic projection we 

 have 



lcja= K sec xj/ 



Hence 



c^ + d' = l + K^sec^}P 



cd=K 

 or 



By means of these formulas the semiaxes could be 

 computed for any point on a continuous map of the 

 sphere or of the ellipsoid if it is desired to take into 

 account the eccentricity of the generating ellipse. As a 

 go6d approximation for projections extending no farther 

 from the central meridian than is usually the case, we 

 may take 



c= K sec ^ = itm 



d = l. 



The effect of this approximation becomes barely perceptible 

 in the third place of decimals for X = 45°, so that the approx- 

 imation is exceedingly good for projections of less extent in 

 longitude. 



With this approximation for the semiaxes it only remains 

 to determine the angles through which the axes of coordi- 

 nates should be turned to make them coincide with the 

 directions of the axes of the ellipse. The angle through 

 which the axes must be turned to make the x axis be tan- 

 gent to the parallel at the point we shall denote by ^; its 

 value is given by the formula 



^ = X sin <p. 



