. . j' for determining the difference of meridians , &c. 125 



In appreciating the weights to be attrihuted tp; these several 

 results, it is obvious that the numbers of corresponding 

 observations at each pair of stations, and of transits at the 

 observatories, as it essentially influences the probable accu- 

 racy of the mean comparison of their timekeepers must be 

 the elements of all fair estimations. If corresponding ob- 

 servations at any station be wanting, the weight is evidently 

 r\othing; so that calling x,y, z, the numbers of correspond- 

 ing observations at A and B, at B and G, and at C and Z M 

 respectively, x xyxz must necessarily be a multiplier of the 

 function expressing the joint weight of the whole. But if 

 the number of observations at any one station, or at all, be j.,,^,, 

 infinitely multiplied, the weight is clearly not infinite. If at 

 all the stations, it would afford only such a degree of evi- 

 dence as a perfect comparison of the clocks would give, 

 which is but a relative certainty, after all, and may be 

 denoted by unity. In like manner, if the observations at any 

 one pair of stations be infinitely multiplied, the result is still 

 open to all the errors of imperfect observations at the rest, 

 so that unity will in like manner be the maximum of the 

 coefficient depending on any separate set. The function 



ja y z 



is the simplest which satisfies these conditions, each factor 

 vanishing when its variable is 0, and becoming unity when 

 infinite. The same reasoning applies to the transit observa- 

 tions by which the clocks are compared with the stars, so 

 that calling T and t the number of transit observations taken 

 at each, by which the clock's errors are obtained, the function 

 expressive of the weight of any night's observations will be 



