THEORY OF POLYCONIC PROJECTIONS. 



119 



CONVENTIONAL POLYCONIC PROJECTIONS. 



There is a class of projections that are not strictly equal- 

 area, but which have the property that they preserve the 

 area of the zones between the parallels and that of the 

 lunes between the meridians. Any equal-area projection 

 possesses this property, but it is not conversely true that 

 any projection possessing this property is also an equal- 

 area projection. Tissot calls projections of this class 

 atractozonic. It can be rigidly proved that no rectangular 

 polyconic projection can be an equal-area projection. We 

 can, however, have an atractozonic projection in the 

 polyconic class that « 



also has circular ^ 



meridians forming a 

 rectangular net with 

 the circular parallels. 



In those that we 

 shall study first we 

 shall take the 

 straight-line paral- 

 lel of the map to 

 represent the Equa- 

 tor, and the circum- 

 ference described 

 upon the line of 

 poles of the map as 

 diameter to repre- 

 sent the meridian the 

 longitude of which is 

 90°, reckoned from 

 the central meridian 

 or the line of poles. 

 We shall determine 

 ^' as a function of ^^ 

 in such a manner that, in the hemisphere limited by this 

 meridian, the area of the half zone comprised between any 

 two parallels will be preserved, and we shall determine X' 

 as a function of X, so that the area of the lime formed by 

 any two meridians may be preserved. The equal-area 

 projections not only have the zones and lunes equal, but 

 also in them the meridians of the earth and those of the 

 map, respectively, divide each zone into proportional parts. 

 This latter property is not foimd in the atractozonic 

 projections. 



In figure 36 we shall suppose the radius OA or OP equal 

 to ^, so that the hemisphere and the circle which serves-as 

 its projection are equivalent, since the radius of the globe 



Fig. 35.— Geometrical relations of atractozonic projections. 



