for determining the difference of meridians , &c, 109 



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 18th, 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 j3 the rate 

 or sidereal time of the chronometer at the 2d station (B), 

 /3 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 — jG (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— | B— /3 (A— E) jor A— B+ iQ(A-E) 

 will be the difference of the clock and chronometer reduced 



