Page:A Voyage to Terra Australis Volume 1.djvu/495

The time keepers.] any given time, is the sum of all the terms of a series in arithmetic progression; of which the daily acceleration, turned into longitude, is both the first term and common difference; and the days elapsed form the number of terms. Therefore,

Daily accel. × 15, × number of days + 1, × half number of days, = the difference of longitude in seconds.

The differences thus obtained have been applied to the errors in longitude on arriving, as given by the rate before found; and the result is the error which remains with rate accelerated.

Had the time keepers been so far perfect machines, as that the change in their rates had been altogether gradual and uniform; and were the stations, as fixed by the lunar distances, perfectly exact; then the error on arriving, with rate accelerated, would have been 0′ 0″ in every case. But since neither the one nor the other may be perfect, and yet it is necessary to take one of them as being so, the difference which remains is considered to be the deviation of the time keepers from an uniformly accelerating rate; and this I call their supplemental error. It is generally impossible to fix upon the manner how, or time when this error was contracted; and therefore it is distributed equally throughout the whole passage, day by day, according to the quantities marked in the last column of the table.

To recapitulate briefly: The longitudes of places given in this first Book of the voyage, and used in the construction of the charts of the South Coast, are those resulting from the time keepers, corrected as above for the acceleration of their mean rate and the supplemental error. The exceptions are, King George's Sound, Port Lincoln, and Port Jackson, which are fixed by the lunar distances; and with the position of Simon's Town, Cape of Good Hope, constitute the basis of this chronometrical fabric.