106 Mr. Herschel’s account of a series of observations 
chronometer ; then, if the observations were perfect, the 
difference of the clock at A, and the chronometer at B, would 
become exactly known. Let this be denoted by A — B. 
A short time after, let a signal be made at b , and observed by 
the chronometers at B and C, whose difference (which we 
will in like manner denote by B — C,) becomes thus precisely 
known at the time of making the signal. In the same manner 
may the difference C — Z of the chronometer at C and the 
sidereal clock at Z be known at the moment of explosion of 
a signal at c ; and so on, if there be more intermediate 
stations. 
Now, the clocks at A and Z being all along supposed to 
keep strict sidereal time, if the watches at B, C, did the same, 
it is manifest that the difference between any two of them 
determined at one moment would be the same at every 
other ; and therefore the intervals elapsed between the 
signals would be out of the question, and the observations 
might all be regarded as simultaneous ; so that the sum of 
the differences (A — B)-}-(B — C) -j- (C — Z) = A — Z 
would express strictly the difference of the true sidereal times 
at the extreme points, i. e. their difference of longitudes ex- 
pressed in time, without any further calculation or reduction. 
It is equally evident that, whatever be the rates of the 
watches, if the intervals elapsed between the signals were 
infinitely small, so as to reduce their gain or loss in these 
times to nothing, the same would hold good. Since this 
however cannot be the case, it is obvious that the difference 
of longitudes so obtained will be affected by the rates of the 
watches and the intervals of the signals, which must accord- 
