375 HYDROGRAPHIC MANUAL PaGE 294 



be computed from its azimuth and distance (see a(l) above), and from that i)oint the other gradua- 

 tions may be plotted. 



(3) The construction of the arcs is accomplished as in the three-arm protractor method by setting 

 on the protractor the angle corresponding to 180°-(x — x') and making the two arms pass through 

 corresponding graduations on the auxiliary lines. The center of the protractor will be a point on the 

 arc sought. If portions of the arc to be drawn are outside the auxiliary straight lines then the angle 

 set on the protractor will be (x— x'), that is, the angle between the two auxiliary lines. (See 3741f (3).) 



It is to bo noted, of course, that in this method it is not absohitely necessary that 

 the arcs be drawn on the sheet as is the case in 3741. The vessel's position may be 

 plotted by swinging the loci of the observed angles. This may be advantageous in 

 certain field investigations or in plotting the smooth sheet. 



375. Modifications of the General Case 



The problem is simplified where the Line of Centers is on the sheet or where one 

 or both the stations are on the sheet. 



3751. Both Stations off the Sheet — Line of Centers on the Sheet 



By referring to the layout sheet, select three points, a suitable distance apart, within the limRs 

 of the survey sheet being prepared and close to the perpendicular bisector of the line joining two of 

 the control points. Their distances to D (fig. 62) are roughly scaled from the layout sheet and dis- 

 tances chosen to the nearest 1,000 meters. Using Form 27, Position Computation Third-Order Tri- 

 angulation, the geographic positions of the selected points are computed from point D along azimuth 

 DB— 90° (fig. 62). A straight line is drawn through these points for the entire length of the survey 

 sheet. This is the Line of Centers. Where only a portion of it falls within the limits of the large- 

 scale sheet, it can be extended some distance beyond and onto anotlier strip of paper by means of a 

 straightedge; or it may be feasible to construct a projection to include the entire Line of Centers and 

 then, after the arcs are constructed, cut off the unnecessary portion. If the latter method is used, 

 points near the extreme limits of the line sliould be computed. 



Distances along the Line of Centers and the radii for the various arcs to be drawn on the sheet 

 are computed as described in 37416(3), These distances are plotted from the nearest computed point, 

 so that practically all distortion along the Line of Centers is eliminated. 



Select a point on one of the arcs near the center of the work and compute its geographic position. 

 This is readily done from the corresponding computed center, the computed radius, and an assumed 

 azimuth. A comparison of the scaled and computed radii will determine the distortion correction 

 (if any) to be applied to the rest of the computed radii. 



The required arcs are drawn by any of the methods described in 3741 c. 



•Under special conditions computations of the radii may be eliminated even where both the stations 

 are off the sheet. For example, if points A and A' (fig. 62) on the perpendicular bisector fall within 

 the sheet, then the locus of the angle a can be plotted from the relationship given in equation (3) in 

 371, namely, that the radius of the locus of angle a is equal to the distance from A to A' on the Line 

 of Centers. 



3752. One Station on the Sheet — Line of Centers on the Sheet 



This is a modification of the above case. Since one of the control stations lies 

 within the large-scale sheet, the radii for the various arcs need not be computed but 

 can be scaled graphically from the plotted centers and the plotted station. 



3753. One Station on the Sheet — Line of Centers off the Sheet 



This condition is best treated by applying the principle of the "Auxiliary Straight- 

 Line Method" (see 3742). The same procedure is followed as described there, except 

 that only one auxiliary line is required and it is drawn from the station that falls off the 

 large-scale sheet. The auxiliary straight line is plotted on the large-scale sheet and 

 graduated as before. The arcs are drawn with a three-arm protractor by setting off 

 the angle x (fig. 67) on the protractor and making the two arms pass through the station 



