GENERAL THEORY OF POLYCONIC PROJECTIONS. 



By Oscar S. Adams, 

 Geodetic Computer, U. S. Coast and Geodetic Survey. 



DETERMINATION OF ELLIPSOIDAL EXPRESSIONS. 



In the consideration of the subject of map construction, 

 the initial question to be decided is the manner in which 

 the meridians and parallels are to be represented in an or- 

 derly way upon the plane surface of the map. This is done 

 by the adoption of some mathematical expression that 

 determines a one-to-one relation between tne meridians^ 

 and parallels and their corresponding curves in the plane. 

 In the consideration of this determination, the earth can be 

 looked upon either as a sphere or as an ellipsoid of revolution. 

 When especial accuracy is desired, the eccentricity must be 

 taken into account. It the formulas are determined for the 

 ellipsoid, they can be reduced to those for the sphere by 

 setting the expression for the eccentricity equal to zero. 

 Since the ellipsoidal form is to be taken as the basis of 

 most of the following discussions, a preliminary determi- 

 nation of the necessary lines will be given. 



In figure 1 let EPS represent a quadrant of the generat- 

 ing ellipse. P and P' are contiguous points; PK is the 

 normal at P and P' K the same at P'. If the equa^tion of 

 the elHpse be given in the parametric form 



x = a cos yp 



y = 'b sin ^, 



a will represent the equatorial radius or the semimajor axis, 

 and h the polar radius or semiminor axis ; ^ is the eccentric 

 angle as indicated in figure 1 If (^ is the latitude of the 

 point P, it will be seen that 



tan^=-^, 



but 



dx= —a sin ^ dyj/ 



dy= h cos \p drp. 



