THEORY OF ERRORS. 



I. Laws of Error. 



The theory of errors is that branch of mathematical science which considers the 

 nature and extent of errors in derived quantities due to errors in the data on 

 which such quantities depend. A law of error is a relation between the magni- 

 tude of an error and the probability of its occurrence. The simplest case of a 

 law of error is that in which all possible errors (in the system of errors) are 

 equally likely to occur. An example of such a case is had in the errors of 

 tabular logarithms, natural trigonometric functions, etc. ; all errors from zero to 

 a half unit in the last tabular place being equally likely to occur. 



When quantities subject to errors following simple laws are combined in any 

 manner, the law of error of the quantity resulting from the combination is in 

 general more complex than that of either component. 



Let e denote the magnitude of any error in a system of errors whose law of 

 error is defined by ^(e). Then if c vary continuously the probability of its 

 occurrence will be expressed by <^(e)^e. If e vary continuously between equal 

 positive and negative limits whose magnitude is a, the sum of all the probabili- 

 ties 4){e)de must be unity, or 



+ a 



— a 



For the case of tabular logarithms, etc., alluded to above, ^(e) = r, a constant 

 whose value is 1/(2 a) = i, since a = 0.5. 



For the case of a logarithm interpolated between two consecutive tabular 

 values, by the formula v=iVi-\- (vo — 7'i) /=: 7\ (i — /) -{- r.y t, where v^ and 

 ?'2 are the tabular values, and / the interval between v^ and the derived value 

 V, ^(e) has the following remarkable forms when the extra decimals (practically 

 the first of them) in (z', — v^ t are retained : — 



<^(e) = - "* for values of e between — ^ and — (i — t), 

 \i t) t 



= _ for values of € between — (i — ^ and + (i ~ 0> (^) 



= . '_ /\ f for values of e between + (^ — ^"^ 4" i* 



