Page 293 hydrography 3742 



drawn with the aid of a three-arm protractor. The method is adaptable for use without 

 the advance drawing of the arcs on the large-scale sheet since, by means of the graduated 

 auxiliary lines, the position of the vessel may be plotted from the loci of the two observed 

 angles. This may be advantageous where limited offshore investigations are required, 

 and cannot be done with the graduated perpendicular method. 



a. Preliminary construction 



(1) On the layout sheet (fig. 66), draw two auxiliary straight lines ME and ML, through one of 

 the stations to be observed on, making angles x and x' with line MP and located on the side of the line 

 where the arcs are to be constructed. The azimuths of these lines are scaled to the nearest degree. 



(2) From P, the other station to be observed on, erect perpendiculars to the two auxiliary lines, 

 intersecting them at points F and C. Select three well-distributed points G, H, and A' on each auxiliary 

 line and scale their distances to the nearest 1,000 meters from points F and C. 



b. Computations 



(1) With the scaled azimuths and distances compute the geographic positions of F and C and 

 of the three selected points (G, H, and K) on each line. 



(2) Compute the distances PF and PC, and from equation (4), d=X-^cot a, and the table of 



natural half-cotangents in 964, compute the distances along FE and CL for various angles a to cover 

 the entire range of curves to be drawn, a in the formula being equal to 2PF and 2PC. These gradua- 

 tions are points where the auxiliary straight lines intersect the various arcs passing through M and P. 



The truth of this proposition is demonstrable from a consideration of figure 67. 



In the figure, DO is the Line of Centers for stations M and P, and ME is an auxDiary straight line making an angle x with MP. 

 O is the center of the circle which is the locus of all points where angle a is observable and which intersects the auxiliary line at point 

 E. If, from station P, a perpendicular PP' is drawn to ME makmg FP' = FP, then ME is the Line of Centers for circles passing 

 through P and P' and by equation (1) in 371, 



<^") 



Therefore, graduations along the line ME for various angles a will be points on the arcs of corresponding angular values and which 

 pass through stations M and P. 



c. Construction of arcs 



(1) On the large-scale sheet plot the three selected points G, H, and K on each of the auxiliary 

 lines from their computed geographic positions (b (1) above), and draw straight lines through them. 



(2) Plot the graduations on the auxiliary straight lines from the values obtained in b (2) above. 

 To do this the geographic position of one of the graduations on each of the auxiliary lines must first 



