188 hersey: criterion for best magnitudes 



tity x, then the function 4>(x) is defined by the statement 

 Ax = <t>{x)-e, where the constant e represents this error at some 

 designated region of x, such as the zero point of a scale. On the 

 other hand, if Ay be the resultant error, then the function Q(y) is 

 defined by the understanding that it is Q(y) • Ay that we wish 

 to make a minimum by our choice of best magnitudes for the x's. 

 Granting that the resultant error is 



*V= 2 



dy 



-f- • Ax T 



^-* dx r 



r = l * 



it is seen that to make Q(y)- Ay a minimum, requires a minimum 

 for 



r = n 



r=l r 



Differentiating by any particular x such as x k will lead to the fol- 

 lowing condition for a maximum or minimum if we simplify the 

 notation by omitting functional parentheses and by letting p 

 stand for df/dx, and if we cast out as physically zero all deriva- 

 tives of 4> r by x k except that of k itself: 



sJr2**ft+*l i *+-IN*5? r - (1) 



9 dx K f* dx K f* dx K 



This equation is the type for a series of n simultaneous equations, 

 the group constituting the criterion. 



Those who are partial to the theory of probability will remark 



that we have taken for Ay the actual error > y- Ax and not 



the (smaller) probable error 



The treatment of 



best magnitudes differs, however, from the greater part of the 

 theory of errors in that it deals with the prevention and not with 

 the computation of errors, and therefore must concern itself with 

 the case when the errors do reinforce each other by pure addition. 

 Nevertheless, the criterion for minimum probable error can be 

 obtained from the other by replacing e, 0, and p by e 2 , cj> 2 , and p 2 , 

 and multiplying the first summation by 2. When there is but 



