hersey: criterion for best magnitudes 189 



one independent variable to consider, the distinction between 

 actual and probable error vanishes. 



Now from (1) by substituting n = 2, Xl = x, x 2 = z, p, = />, 

 Pi = q, 6 = l/f, and = 1, we have the important special case 

 of two independent variables with constant errors e x and « z , a 

 minimum fractional error Ay/y being desired in the result. The 

 criterion reduces to the single equation 



<»-«>•*-(£-©•/■ » 



where R denotes (pe* + qe z ). 



Finally for one independent variable, if we denote differentia- 

 tion by primes and put p = f, (1) becomes: 



f e' + /' 0' e +/" 00=0 (3) 



This breaks up into eight special cases upon assigning to 6 

 the fundamental values 1 and l/f while is restricted to 1 and x. 

 The four most important in practice are the following: 



1. When seeking a minimum absolute error, 6 = 1, conse- 

 quently the criterion becomes: 



/V +/"</, =0 (4) 



2. When seeking a minimum fractional error 6 = l/f. 



■ -*~7 r () 



3. When seeking minimum absolute error with constant error 

 in x, 6 = 1 and = 1, therefore 



/" = (6) 



4. When seeking minimum fractional error with constant 

 error in x, 6 = 1// and <p = 1, so that 



• /' = \Tr ~ 



To fix in mind the meaning of the criterion, two examples, in 



themselves sufficiently simple to be verified by trial, may be 

 added, — 



I. Find best resistance for heating coil if power input // = 

 x-z is deduced from readings of 15 a. ammeter with error e, and 



