109 
for determining the difference of meridians, &c. 
to reject all such broken series, the observations of a whole 
night might easily be thrown away, though capable of 
affording a result quite as good as any other. Such a case 
actually occurs in the observations of the 18 th, where no 
complete transmission of any one signal from end to end of 
the line took place, yet the mean result of that night’s obser- 
vations deviates less than two-tenths of a second from the 
result finally adopted as the truth. 
The most advantageous way of employing such a broken 
series of observations as we have described is not at once 
obvious. It may depend on circumstances too nice for cal- 
culation, and which can be felt only by the observers them- 
selves. The fairest however, and that which by employing 
all the observations according to one uniform rule leaves 
nothing to partiality, seems to me to be the following. 
Let A be the time marked by the sidereal clock at the first 
extreme station A, then calling E the time marked by the 
same clock at any assumed arbitrary epoch, A — E will denote 
the sidereal time elapsed since that epoch. Call /3 the rate 
or sidereal time of the chronometer at the 2 d station (B), 
being supposed negative when the chronometer loses, (as 
for instance when it shows mean time). At the same moment 
that the clock at A marks A, let this chronometer mark B, 
then, since /3(A — E) is the quantity it has gained, since the 
epochs, B — jQ (A — E) must be the time it would have indi- 
cated, if instead of gaining or losing, it had kept true sidereal 
time since the epoch. Consequently (the clock being sup- 
posed to have no rate) A — j B — /3 (A — E) j or A — B+ 0(A-E) 
will be the difference of the clock and chronometer reduced 
