Ixxxiv THEORY OK ERRORS. 



It thus appears that </)(e) in this case is represented by the upper base and the 

 two sides of a trapezoid. 



When, as is usually the practice, the quantity (v., — Vi) t is rounded to the 

 nearest unit of the last tabular place, c/)(€) becomes more complex, but is still 

 represented by a series of straight lines. It is worthy of remark that the latter 

 species of interpolated value is considerably less precise than the former, wherein 

 an additional figure beyond the last tabular place is retained. 



When an infinite number of infinitesimal errors, each subject to the law of con- 

 stant probability and each as likely to be positive as negative, are combined by 

 addition, the law of the resultant error is of remarkable simplicity and generality. 

 It is expressed by 



<^«=^^^=^' (2) 



where e is the Napierian base, tt = 3.14159 -j-, and h is a constant dependent on 

 the relative magnitude of the errors in the system. This is the law of error of 

 least squares. It is the law followed more or less closely by most species 

 of observational errors. Its general use is justified by experience rather than 

 by mathematical deduction. 



a. Probable, mean, and average errors. 



For the purposes of comparison of different systems of errors following the 

 same law, three different terms are in use. These are \\\% probable error* or that 

 error in the system which is as likely to be exceeded as not ; the mean error, or 

 that error which is the square root of the mean of the squares of all errors in the 

 system ; and the average error, which is the average, regardless of sign, of all 

 errors in the system. Denote these errors by Cp, e,„, €„, respectively. Then in all 

 systems in which positive and negative errors of equal magnitude are equally 

 likely to occur, and in which the limits of error are denoted by — a and -j- a, the 

 analytical definitions of the probable, mean, and average errors are : — 



— (j, o -f f,, -\- a 



J'c/>(e) ^e = Jcj^ie) d, r= ^ ^{i) d, = Jc{>(e) de = I 



— a — (p o + ^P 



(3) 



-\- a -\- a 



^\ = fK^) ^' d., e„ = fct>(,) c d.. 



— a —a 



* The reader should observe that the word probable is here used in a specially technical sense. 

 Thus, the probable error is not " the most probable error," nor " the most probable value of the 

 actual error," etc., as commonly interpreted. 



