Suppose by way of explanation that the altitude of a star bearing S. 55° E. is observed simultaneously with that of another star bearing S. 25° E. The longitudes derived by working the time sight of each should be identical, provided the altitudes are the true altitudes, the Greenwich time is without error, and the latitude used is correct. A combination of accuracy, indeed, and one not likely to be experienced often in actual practice. However, a skillful navigator should find no great difficulty these days in always having the correct Greenwich time at hand. There is always, of course, an opportunity for the display of skill in measuring altitudes, refraction particularly being an illusive element and not always easy to detect. But if care has been taken to eliminate the errors as much as possible from the time and the altitude, it is safe to consider any discrepancy between the resulting longitudes as accountable to an error in the dead reckoning latitude.
The method of obtaining the ship’s position from the difference in the longitudes, derived from double or simultaneous observations, was originated by A. C. Johnson, R. N., and its many advantages have for years made it the most popular form among progressive shipmasters. The working of this problem involves the application of a correction to each calculated longitude in such a way as to bring them into agreement. The tables (Bowditch Tables 47 and 48) furnish this correction, which is known as the longitude factor and is symbolized by the letter F. It constitutes the change in longitude due to a change of 1´ in latitude. This quantity changes directly with the change of azimuth of the body; for example, the change in longitude is nil if the change in latitude is made on a due north or south line, and change in longitude increases as the change in latitude is made on lines bearing more and more eastward or westward. So it is necessary in order to obtain these corrections to have the true azimuths of the bodies at the moment of observation to use as an argument in the table of longitude factors. These are readily taken from the Azimuth Tables or diagram using the data furnished by the time sight.
The two longitudes obtained from time sights in which the same dead reckoning latitude is used, lie on the parallel of this latitude, but (unless the two longitudes happen to be coincident) the ship’s position is either north or south of this parallel according to the error existing in the dead reckoning latitude. If the observed azimuth of the body (or bodies) fall within the same quadrant or in opposite quadrants, the correct longitude will be found to the eastward or the westward of both calculated longitudes. This is clearly shown in Fig. 7; both azimuths are between south and east. If the observed azimuth of the body (or bodies) fall in adjacent quadrants say, one between south and east and the other between south and west, the ship’s position will be found between the two calculated or erroneous longitudes. The position of this true longitude is determined by means of the before-mentioned factors. The factor of a longitude is the distance of the true longitude east or west of the meridian passing through the calculated or erroneous longitude, assuming the latitude to be in error 1´. The moment of this factor, it will be seen, depends on the azimuth of the body, which in turn determines the direction of the position line.
Fig. 7.
The combination of the two factors, by adding if the bodies are in the same or opposite quadrants or vice versa, is the combined error in difference of longitude due to 1´ of error in latitude. It now becomes a matter of proportion by which to obtain the error in the dead-reckoning latitude. As the combined error in difference of longitude for 1´ of latitude, is to 1´ of latitude, so is the difference between the two calculated longitudes, to the error in latitude.
The longitude factors are based upon an error of 1´, so if the error is more than 1´ it becomes necessary to multiply the factor by the error in order to obtain the correction to the calculated or erroneous longitude.
Fig. 8.
An altitude may be taken of any body and after a suitable change in bearing has taken place (not less than 30°) a second altitude may be taken and the first longitude advanced for the run during the interval to the parallel of the latitude by dead reckoning at the time of second sight.