MATHEMATICS OF PAPER LOCATION: 

 r RECTANGULAR CO-ORDINATES. The 

 chief points of the preliminary, surveyed by any 

 method or conrbination of methods, have been 

 plotted by rectangular co-ordinates, and our map 

 shows points Pi, Pa, Ps, Pt, PB, P and PT. The co- 

 ordinates of each point are written by the point, 

 the value of the abscissa first. The engineer has 

 drawn in pencil on the map the location Pi, PCi, 

 PTi, PC 2 , PT 2 , PCs. PCi and PTi are respectively 

 the beginning and end of a 6-degree curve; and 

 PCs and PTa are the beginning. and end respec- 

 tively of an 8-degree curve. PCs is the beginning 

 of a curve, and is considered the end of this loca- 

 tion. It will be noticed that each curve is num- 

 bered, and that each letter standing for a curve 

 element is given for a subscript the number of 

 its curve; e. g., the central angle of curve No. 2 is 

 At. All dimensions which appear on the map at 

 this time are put in parentheses on our figure in 

 order to distinguish them from the quantities 

 computed in the work of getting out the location 

 notes, which are written without parentheses. 

 We have nothing to do here with the question of 

 the proper place for the location line on the map: 

 that is an engineering problem, while our work Is 

 to take the location line as we find it, and get out 

 the notes for transferring it to the ground. Hav- 

 ing no need of the topography in what follows, It 

 has been omitted from our map. The distances 

 have been computed here to the nearest foot only, 

 although it is customary to consider tenths of a 

 foot. Of course, in any given location the limit of 

 error in field work should control the precision of 

 computation. To save space there have been 

 omitted from Fig. 1 the station numbering of the 

 preliminary and of the location. 



1. PRODUCE THE TANGENTS to intersect 

 at Vi and Vs. 



2. SCALE THE CO-ORDINATES of the initial 

 point, the final point, and of all the points of tan- 

 gent intersection. In our case, the initial roint of 

 the location is at the origin of co-ordinates, and 

 coincides with the Initial point of the preliminary. 

 The co-ordinates of Vi and Vz and PCs are written 

 by the points on the map as they are scaled off, 

 x first and y second. 



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