THEORY OF POLYCONIO PROJECTIONS. 129 



This projection is sometimes called the stereographic pro- 

 jection with modified meridian. 



NONRECTANGULAR CIRCULAR PROJECTIONS. 



Let us always suppose that to each point of the globe 

 there corresponds one point of the map, and only one, so 

 that the circumferences which serve for the projections of 

 the meridians all pass through two points P and P' in 

 figure 36, which are the projections of the two poles. 

 Let APA'P^ be the circumference described upon PP' as 

 diameter, its center, A A' the diameter perpendicular 

 to PP', TJDTJ' the projection of the parallel of latitude ^p 

 or of colatitude jp, B the point in the prolongation of PP^ 

 which serves as the center for this projected parallel, V 

 the middle point of the chord UU^ common to the two 

 circumferences APAT' and UDW. Further, let PGP' 

 be the projection of the meridian of longitude X, reckoned 

 from the central meiidian projected into the line PP' and 

 let T be the center of the circumference PGP' . Let us 

 continue to define this last by the angle X' at which it 

 intersects PP' , which is equal to the angle OTP, so that 

 in the triangle TP we have, as formerly, on taking OP 

 as unity and on denoting by R and 5, respectively, the 

 radius TP and the distance OT, 



P/=cosecX', 5' = cotX', R^-^^^j^ 



As to the projection TJBJJ' of the parallel, we can define 

 it by the two lengths r and s, as we have done up to this 

 time, or by the two angles which the sides of the triangle 

 OSU make with each other. Let us call the angle SOU, 

 p'; its complement, (p'; the angle OSU, €; and, finally, 

 let 7 denote the angle which one of the radii OU and SU 

 makes with the prolongation of the other. vSince we have 

 0U=1, the triangjle OSU is determined by two of the 



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