154 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
an instructive error is 0; the antecedent probability of an unin- 
structive error is always greater than 0; and the observation should 
certainly be rejected. But since the theory of least squares allows 
no limit whatever to the possible magnitude of instructive errors, 
such rejection involves the admission that the method of least 
squares is not applicable to the case. When an observation involves 
a merely suspicious error, which is neither so large that instructive- 
ness is impossible nor so small as to pass without question, it would 
seem reasonable that the observation should be weighted according 
to the relative magnitudes of the two antecedent probabilities 
which I have mentioned; but this can never be determined with 
any approach to mathematical precision. 
In order to make this matter clear, let us suppose for example 
that ninety-nine observations of equal weight and known to be free 
from uninstructive error are separately written on as many cards; 
that the number 25 is arbitrarily written on a similar card; that 
these hundred cards are thoroughly shuffled; and that ten cards 
being then drawn at random, the following numbers appear on 
them: 15, 18, 14, 25, 17, 16, 15, 18, 16, 17. Let it be required to 
determine from these data, according to the theory of least squares, 
the probability that the number 25 on the fourth card drawn is the 
record of an observation. Here the antecedent probability of an 
uninstructive error is by hypothesis equal to 1-10. 
I commence by assuming a value of the required probability, 
and weight the doubtful observation accordingly. I then proceed 
in the ordinary method and determine an approximation to the 
antecedent probability of the occurrence of a genuine observation 
giving the value 25 by integrating <= fe — © dt between the 
limits corresponding to 24.5 and 25.5, since the observations are 
taken to the nearest unit. This integral is the antecedent proba- 
bility of an instructive error of the given magnitude, tainted with 
the incorrectness of the assumption with which I began. Call this 
P 
to +P 
it agrees with my original assumption, the problem is solved. If it 
does not agree, I have data for a better assumption according to the 
well-known method of trial and error. After a few repetitions of 
the process, as I have found by experiment, an assumption can be 
made that will be verified by agreement with the result. 
integral p. Then is the resulting required probability. If 
