372 HYDROGRAPHIC MANUAL PaGE 288 



Equations (1) and (2) therefore become, 



d=X~cota. (4) 



c=X-x cosec a. (5) 



To facilitate the computations there is included in 964 a table of natural half-cosecants and half- 

 cotangents for angles from 1° to 90°. For larger angles the functions can be derived from this table 

 from the relationship, /(a) =/(180°— «). And since the cotangent of an angle larger than 90° is 

 negative, the d distances along the Line of Centers would ba measured above the line BC (fig. 62). 



The use of the table and formulas is. illustrated by the following example: 



The distance between two control stations is 5,000 meters, the scale of the sheet is 1:40,000, and let a be 4°, then substituting in 

 equations (4) and (5): 



d=-X5,000X7. 1503=8,937.9 meters measured on a 1:10,000 scale meter bar 



4 



and c=-X5,000X7.1678=8,959.8 meters measured on a 1:10,000 scale meter bar. 

 4 



A table of the necessary logarithmic functions should be prepared in advance if their use is 

 preferred. 



372. Preparing Circle Sheets 



The problem of preparing circle sheets consists generally in determining the 

 centers and radii of the arcs and drawing the arcs on the sheet. The method to be 

 followed in a given case depends on the relative positions of the two control stations 

 of any system of loci and the large-scale sheet. Five cases may arise. They are: 



Case 1. Both stations are off the sheet and the Line of Centers is off the sheet. (See 374.) 



Case 2. Both stations are off the sheet and the Line of Centers is on the sheet. (See 375L) 



Case 3. One station is on the sheet and the Line of Centers is on the sheet. (See 3752.) 



Case 4. One station is on the sheet and the Line of Centers is off the sheet. (See 3753.) 



Case 5. Both stations are on the sheet and the Line of Centers is on the sheet. (See 3754.) 



Case 1 may be considered the general case and the other four cases as modifications 

 oj the general case and they are so. treated in this Manual. 



373. Preliminary Layout 



Irrespective of the conditions encountered or of the method used in preparing 

 the circle sheets, a preliminary layout must first be made on a chart of the locality on 

 which the control stations and the limits of the various sheets to be used are accurately 

 plotted (figs. 63 and 66). Tliis should be done prior to any computation or plotting. 

 The layout is needed in selecting the control stations to be used, if there are more 

 than tliree to choose from, and as a guide in determining the angles to be computed. 



It may be necessary to traverse with the vessel around the area to be surveyed to 

 select the best fix available. The importance of this selection cannot be too strongly 

 emphasized. Signals most likely to be visible under all conditions should be selected 

 as this will often reduce the number of systems of arcs on a sheet and reduce confusion 

 in plotting. The selection of stations is not limited to those constituting a three-point 

 fix, but any two pairs that give good intersecting arcs may be used. 



374. General Case 



BOTH stations OFF THE SHEET LINE OF CENTERS OFF THE SHEET 



There are two methods of solving the general case: the graduated perpendicular 

 method, and the auxiliary straight-line method. Both are described and the hydrog- 

 rapher should select the one which best suits his particular conditions. 



