96 U. S. COAST AND GEODETIC SURVEY. 



GENERAL STUDY OF DOUBLE CIRCULAR PROJECTIONS. 



In order to enter upon some points not yet discussed, 

 we shall study in general those projections in which the 

 meridians are represented by a system of circles passing 

 through two common points which form the poles of the 

 projection and in which the parallels are represented by 

 a system of curves orthogonal to the meridians. The 

 centers of the circles forming the meridians will all lie 

 upon the perpendicular bisector of the common chord 

 which forms the line joining the poles of the projection. 

 The tangents drawn to the various circumferences from 

 any point of the prolongation of the common chord are 

 equal, since they are in each case a mean proportional 

 between the same secant and the external segment of the 

 same. If from this point as center, with a radius equal 

 to one* of these tangents, we describe a circle, it will inter- 

 sect aU the circular arcs representing the meridians at 

 right angles. We thus see that the orthogonal trajec- 

 tories of the meridians of the map — that is, the parallels — 

 are also circumferences, so that they belong to the poly- 

 conic projections. The locus of centers of the parallels 

 is a straight line passing through the projections of the 

 two poles and perpendicular to the locus of centers of 

 the meridians. 



Every point of either prolongation of the line ot poles of 

 the map can be considered as the center of the projection 

 of one of the parallels, and the radius of this projection is 

 then equal to the tangent drawn through the point in 

 question to one of the meridians of the map; for example, 

 to the circumference described upon the line of poles as 

 diameter. Reciprocally, if in a projection with orthogonal 

 curves the parallels are circumferences having their centers 

 upon the prolongations of one of the diameters of a given 

 circumference and as radii the tangents drawn from the 

 various centers to this circumference, the meridians will 

 also be circumferences which pass through the two extrem- 

 ities of the given diameter. This wiU not be true if the 

 radii of the parallels are determined by any other condition 

 than the one mentioned. The rectangular polyconic pro- 

 jection of the English War Office, already discussed, fur- 

 nishes an example of an othogonal projection in which the 

 parallels, but not the meridians, are circumferences. 



The properties which we have just pointed out are not 

 the only ones which we can extend from the stereographic 

 projection to all conformal projections with circular 

 meridians and from these to projections with circular 



