The time keepers.'] APPENDIX. 
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, x number of days \ 1, x 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 per fee: 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 o' o" 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 there- 
fore 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 'hose resulting from the time keepers, cor- 
rected as above for the acceleration of their mean rate and the sup- 
plemental 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, consti- 
tute the basis of this chronometrical fabric. 
END OF VOL. I. 
London : Printed by W. Bulmer and Co. 
Cleveland Row, St. James's. 
