258 
PROFESSOE K. PEARSON ON THE MATHEMATICAL THEORY 
probable errors of the differences, i.e., there is a real individuality in observation 
which manifests itself in the personal equation. 
But the fluctuations in the personal equation are significant too, and they cannot 
offhand he attributed to anything like betterment with practice, or decadence with 
fatigue. 
So long as the variations in the constants of an experimental series can be shown 
to be within the errors of random sampling we feel on safe ground ; we know the 
number of experiments required to obtain a result with any required degree of 
accuracy. On the other hand, when we find significant fluctuations in the personal 
equation depending on the influence of immediate atmosphere, it becomes all the 
more important to show in each individual investigation that the personal equation 
itself is insignificant. Let me illustrate this point. A physicist makes twenty or 
thirty measurements of a quantity, say by aid of a bright line moving across a 
scale. He gives the mean value m of the result and also what he terms its probable 
error e. Now the use of this probable error 1 take it to be this. If the same experi¬ 
ments were to be repeated by the same man the same number of times with the 
mean result m', then we should expect to find m — m not a large multiple of the 
probable error of the difference = \/(^) ^ gives us a test of the 
closeness with vliich the result will repeat itself on repetition of the experiments. 
But the wliole foundation of this statement is the hypothesis that the twenty or 
thirty experiments dealt with are a random sampling of all possible experiments 
that might be made. Now the variability in the results of the individual 
experiments includes the variability of personal error, and the hypothesis supposes 
that the personal errors are a random sampling of the observer’s personal errors. 
Our investigations seem to indicate that the personal errors are far from being a 
random sampling but depend in some subtle manner on the influence of immediate 
atmosphere. Hence, unless it can be shown that the latter influence is small as 
compared with other sources of error in the measurement under consideration, the 
mere calculation of the probalfle error is l3y no means a security for the same 
observer reaching the same result on repeating the original series of experiments. 
We, of course, for both series selected exj^eriments in which the personal error 
would be large,and accordingly could be easily dealt with. But the division of 
scale lengths by the eye and the estimated position of a bright line are fundamental 
in many types of pliysical observation. Further, large errors are for theoretical 
purposes quite as good as, for practical purposes much better than, small, when we 
wish to olDtain an answer to the question ; Are the fluctuations in personal equation 
merely the result of random sampling, or are they due to the influence of immediate 
atmosphere ? 
So important is it to realise that these fluctuations are not due to random sampling, 
* As a matter of fact only at a maximum in dividing a line and in determining the position 
of a bright line lietAveen two scale niark.s. 
