98 U. S. COAST AND GEODETIC SURVEY. 



U'A'Uj^, TJA^ also bisects the angle V'UV^; therefore, the 

 three points Z7, D, A' lie on a straight line which makes it 

 possible to construct the point Z> without describing the 

 circumference 8 when Z7is given. Since the angles ADA' y 

 DVD' J each, being inscribed in a semicircle, are right 

 angles, the three points A, U, D' also lie on a straight line, 

 which is the bisector of the angle formed by one of the 

 sides of the triangle V'TJV^ vidth the prolongation of the 

 other. 



The angle PUA'j which subtends, upon the circumfer- 

 ence 0, an arcequal to a quarter of the circumference, is 

 equal to the half of a right angle; the same is true of the 

 angle DVF' , which subtends upon the circumference S 

 an arc equal to a quadrant; the two angles are, therefore, 

 equal, and, as two of their sides VA' and VD coincide, the 

 two others, TIP and Z7F', also coincide; that is to say, that 

 the points TJ, P, F' are in a straight line. Since UP' is 

 perpendicular to 77P and TJFto Z7P', the points P', TJ, F 

 are also in a straight line. It follows from this that VD 

 is the bisector of the right angle PZ7P' and VD' of the 

 adjacent angle PVF] therefore, DP :DP'=p'P : D'P' = 

 VP : VP', The projection of each parallel is the locus of 

 the points the distances of which to the projections of the 

 two poles have a given fixed ratio. The lines VP and 

 VP' are in their turn bisectors of the right angles DVD' 

 and DVA; therefore, the ratio of the distances of an^y 

 point of the circumference to the two points D and D' is 

 constant. 



In figure 31 the letters already appearing in figure 30 are 

 employed with the same signification. The semicircum- 

 ference PAP' is the projection of a particular meridian. 

 Let us now consider the projection PMGP' of any meridian. 

 Let T be the center, G and M its intersections with A A' 

 and the circumference S, respectively, and, finally, let G' 

 and M' be the points of intersection of the arc which com- 

 pletes the circumference T with the same two lines, respec- 

 tively. With regard to the two circumferences S and T, 

 we should have to point out the same properties that were 

 pointed out as obtaining between the two circumferences 

 S and 0. It will be sufficient to indicate the following 

 facts: Since J/ lies on the parallel circle which is the locus 

 of points with distances from P and P' in the ratio DP to 

 DP' J the ratio of MP to MP' is the same as that of DP to 

 DP'\ therefore, the line MD is the bisector of the angle 

 PMP', and it should pass through the mid-point G' of the 

 arc PG'P'; then the three points If, Z>, (r' are in a straight 



