THEORY OF POL! CONIC PROJECTIONS. 75 



Therefore, by integration, 



4(«4> 



in whicli the constant of integration may be taken as 

 zero, since the addition of any quantity would only serve 

 to change the point from which s is reckoned. 

 From these results we obtain 



s-{-p = cu 



c 



s — p =- 



u 



or, by multiphcation, 



This equation shows that the circle -^-ith the origin as 

 center, constructed with the radius c, cuts all the parallels 

 at right angles. Any circle drawn through the two points 

 of intersection of this circle and the line of centers of the 

 parallels will also cut the parallels orthogonally, for the 

 tangents drawn to it from any point in this line*^of centers 

 are equal. Therefore, these circles, since they form the 

 orthogonal trajectories of the parallels of the map, are 

 none other than the projections of the meridians. The 

 two common points in the line of centers of the parallels 

 are the poles of the map. 



If, then, we take two arbitrary points to represent the 

 two poles, the meridians of the map will be the arcs of 

 circles which pass through these two points and the 

 parallels will be other arcs of circles having their centers 

 at various points of the prolongation of the hne of poles 

 and each passing through the point of contact of the 

 tangent drawn from the center to any one of the merid- 

 ians ; for example, to the circumference described upon the 

 hne of poles as diameter. 



We have yet to find the expressions for u, p, and s in 

 terms of (p, and that for T (X) in terms of X, by which expres- 

 sions we may be able to tell, in the first series of arcs, 

 the one that corresponds to a given meridian X and, 

 in the second series of arcs, the one that corresponds ta 

 the parallel of latitude (p. 



