The relative bearing of an object is its direction from the ship relative to the ship's head. It is the angle between the fore-and-aft line of the ship and the bearing line of the object, measured clockwise from 000° at the ship's head through 360°. The pelorus can be used for taking relative bearings by setting the 000° graduation of the pelorus card to the lubber's line, then observing the object and reading the card.
Relative bearings are converted to true bearings before they are plotted. This is done by adding their value to the ship's true heading when the relative bearings were taken, subtracting 360° if the sum equals or exceeds that amount.
When selecting objects from which to obtain a fix, the primary consideration is the angle between the bearings. If only two visual bearings are available, the best fix results from two bearings crossing at 90°, in which case an error in either bearing results in minimal error in the plotted fix. As the angle between the objects decreases, a small error in either bearing throws the fix out by an increasing amount. Bearings of objects intersecting at less than 30° should be used only when no other objects are available, and the resulting fix should be regarded with caution.
To check two bearings and to minimize fix error, three or more bearings should always be taken if possible. If three are taken, the optimum angle is 120° (or 60°) between bearings.There is no rule for advancing a terrestrial line of position,I like to keep them no more than 30 minutes.It is pbssible to solve the running fix by trigonometry; two angles are determined by measurement, and the length of the side between them is determined by the ship's run between the bearings. The distance off at the time of the second bearing can be found as can the predicted distance off when the object is abeam. These calculations can be done on any small hand calculator that has trigonometric functions.
Solution by Table 7, Bowditch
It is not necessary to resort to trigonometry to obtain the solution.
I like to use table 7 Bowditch and have had good results,it tabulates both distance off at the second bearing and predicted distance off when abeam, for a run of one mile between relative bearings from 20° on the bow to 30° on the quarter. Since the distance rarely equals exactly one mile, the tabulations are in multipliers or factors, which, when multiplied by the actual run give the distance from the object at the time of the second bearing and the predicted distance at which the object should be passed abeam. I use a format for this kind of a problem,you can click on the link at the top of my site.Arguments for entering Table 7 are arranged across the top and down the left side of each page. The multipliers or factors are arranged in double columns. The left-hand column lists the factors for finding the distance at the time of the second bearing. The right-hand column contains the factors for finding the predicted distance abeam.
Whenever the second bearing is 90° (relative), the two factors are the same. In this case the second bearing is the beam bearing and the element of prediction no longer exists.
In case the second bearing is greater than 90° (relative), the right hand factor no longer gives a predicted distance abeam, but the estimated distance at which the object was passed abeam.
It must be remembered that the ship's heading (SH) may not be the same as the course being made good over the bottom (CMG); for example, the vessel may be "crabbing" slightly, heading a bit into a cross current in order to make good a desired track. In the general cases of this and in the more special cases the angles must be measured with respect to the course being made good over the bottom. If there are any currents influencing the track of the ship with respect to the earth, appropriate corrections must be applied.
There is usually no need to interpolate when using Table 7, even though only the even-numbered relative bearings are given. As a rule it is easy to obtain even-numbered relative bearings if the bearing taker or navigator exercises a little patience.There are special cases which do not require the use of Table 7,they are bow and beam 45 - 90 degree bearings,22 1/2 - 45 degree rule doubling the angle on the bow,30 - 60 case or the 7/8 rule,and 26 1/2 - 45 degree case. I use a format for all these cases and will have these in my link.
