Abstract
When determing a ship's position by the running fix, past explanations have been vague and far from reality, because the nautical distance and its error must bc treated as the functions of a ship's speed and sailing time, and also the error of transferred position line is the function of the time. The author studied this problem from the view-point of the theory of probability. When determing a ship's position by a bearing and a transferred position line, the best time for the second observation is when the probability density of the determined ship's position is at maximum value. To get the values he solved the conditional equation (1') and indicated the results in table 1. The sailing times to meet those terms are shown in table 2. Here the error of transferred position line was treated as 0'.75√t (t: sailing hours) based on the experiment [1] But, when the weather is stormy and the effects of ocean and tidal currents are strong, to transfer the first position line is difficult, and on the other hand, when the sea is calm and the current is feeble, the estimated magnitude is small, and then the error of estimation is naturally considered to be a small amount. From this point of view, he assumed the coefficient to 1'.05 for unfavorable conditions and 0'.45 for the favorable conditions, and calculated the optimum angles, the results are shown in table 3 and they differ little from the values in table 1. However according to the idea advocated by some people that the error of estimation is in proportion to the sailing time, the error is considered to be 0'.53t (s.d), and the calculated optimum angles based on this way of thinking are indicated in table 3. The results differ somewhat from the aforesaid results. Making the second observation at the best time, i.e. when the probability density of the determined ship's position is at maximum value, has a drawback that the chance occurs after the ship is abeam the object. And so, secondly, he studied the determination of a ship's position by use of two transferred position lines and a bearing when a ship is abeam the object. The results gotten by the equation (2) are indicated in table 4. As shown in the table, it is interesting that the best angle of observation is situated between 45° and 47° without regard to V, d, a and α. Against them, if he treats the error of estimation as 0'.53t (s.d), the conditional equation indicated by (3) is formed and the results are shown in table 5.
