125 
for determining the difference of meridians, &c. 
In appreciating the weights to be attributed to 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 
nothing ; so that calling x, y, %, the numbers of correspond- 
ing observations at A and B, at B and C, and at C and Z 
respectively, x xjy x z 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 
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 
x y z 
x+~t X y + i X z + i 
is the simplest which satisfies these conditions, each factor 
vanishing when its variable is o, 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 
