STORY. — A NEW GENERAL THEORY OF ERRORS. 171 



and the probability that the residual of au observation lies between 

 a and b is 



(12) J4,{e)'d^. 



The value of </'(t) for any given value ^ may be called the relative fre- 

 quency of the residual ^, and ^ {$) • d^ is evidently the probability that 

 the residual lies between $ and i + d$ ; that is, ij/ (i) ' d$ is the propor- 

 tion of all the residuals that lie between ^ and ^ -h d^. In calculating 

 the mean powers of the residual by (4), the power of eacli residual is to 

 be multiplied by the number of times it occurs, and the sum of the 

 products thus formed for all the residuals is to be divided by the number 

 of observations ; if we assign all the residuals between | and $ -\- d^ io 

 the value ^ itself (or replace them all by their mean), the multiplier 

 corresponding to this value is proportional to i/^ (^) • d^, while the whole 

 number of observations is proportional to (11). We have then 



f 



(13) p,= fei^(o-d$, 



i 



and, in particular, by (5) and (6), 



I 

 (1^) j'^(0-di = p,= l, 



\ 



(15) Ji^(0-d$ = p,^O, 



i 

 1 



(10) Je^(^)'d^=p,^fj.\ 



i 



Therefore, by (3), (14), and (15), 



X I 



(17) jx-<i,(x)'dx=j($-\-x,)-ij, ($) ■ di = Xo 



£ i 



Because i// (|) is essentially positive for any possible residual $, formula 

 (15) shows that the possible values of $ must include both positive and 

 negative values, so that ^ < 0, < |, and is a possible residual in 

 all cases. 



