of experiments on the pendulum. 77 
vary from .375 to .349 ; and in a greater number of trials the 
errors would approach still nearer to equality. 
Now in order to employ any of these suppositions for the 
purpose of calculation, it is only necessary to compute the 
corresponding mean error, and to make it equal to the actual 
mean error of a great number of observations. Thus, if we 
consider each observation as representing a binary combination 
of counters or constant errors, in which the mean error is - , 
and adding together the differences of the several results from 
the mean, and dividing by their numbers, we find the mean 
error of 100 observations i', we must consider the original 
constant error as equal to 2', which is to be made the unit for 
200 primitive combinations ; and —1 — = .0426 ; and 
the probable error of the mean will be .0426 x 120 = 5.1°. 
For a quaternary combination, if the error, which amounts to 
f, be found i', the unit will be J-', and for n = 400, we have 
.03125 x 5. o". And if we set out with a large number 
m of combinations, the mean error beinar V— — e , the unit 
0 pm 
will be e y/ {pm) = 1, and the probable error of nm trials 
t 
being equal to this unit multiplied by .542 \/~, neglecting 
the very small fraction we have .542 e \/ ( pm ) = 
.542 p y / “ e — -8514 v/7 e ' which, if e be i', and n = 100, 
gives again 5.1". It appears therefore that the supposition, 
respecting the number of combinations representing the scale 
of error, scarcely makes a perceptible difference in the result, 
after the exclusion of the constant error : and that we may 
safely represent the probable error of the mean result of n 
observations, by the expression .85 e being the mean of 
all the actual errors. 
