STORY. — A NEW GENERAL THEORY OF ERRORS. 169 



(7) 



A little consideration will make it clear that no observations, however 

 many, can determine the true value of the observed magnitude, nor, 

 therefore, the errors of the several observations, — they can determine 

 only what is equivalent to their arithmetical mean and the residuals of 

 the observations. The fundamental problem of the theory of errors 

 must, therefore, be modified, and may now be stated to be the determina- 

 tion of the probability that the residual of an observation shall lie 

 between certain limits according to the law of the long run, that is, in 

 general, on the assumption that the distribution of errors over the whole 

 range of possible error tends to follow a certain law, as the number of 

 observations is increased without limit, and that the proportion of all the 

 errors of any (finite) number of observations that fall between given 

 limits is the same as if the number of observations was infinite. It is 

 evident that the distribution of errors among different intervals in definite 

 proportions implies an increase in the breadth of the intervals as the 

 number of observations is diminished. The analytical theory of errors 

 is based upon the supposition that the number of observations is infinite 

 and that the interval for which the proportion of errors is to be deter- 

 mined may be taken to be as narrow as we please. 



In connection with this statement of the fundamental problem it may 

 be said that the determination of the value that may best be assigned to 

 the observed magnitude is not explicitly a part of the fundamental prob- 

 lem, although immediately connected witli it and perhaps the most 

 desirable result of it from the practical point of view, but the criterion 

 for the best value is by no means evident. Perhaps it will be generally 

 considered that the best value is the probable value, that is, the value that 

 is just as likely to be exceeded as not in the long run, so that the prob- 

 ability that the observed value shall fall short of the probable value is \ 

 and the probability that the observed value shall exceed the probable 

 value is \- Certainly the best value cannot be defined as the most prob- 

 able value, that is, the value whose probability is a maximum, because 

 it turns out that the probability is sometimes a maximum for more than 

 one value ; the probability may even have equal maxima for two or more 

 values. 



