JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. I, NOVEMBER 4, 1911. No. 7 



APPLIED MATHEMATICS.— A criterion for best magnitudes 

 in precise measurement. Mayo Dyer Hersey, Bureau of 

 Standards. Communicated by L. A. Fischer. 



In the determination of a quantity y from observations upon 

 the independent quantities X\, x 2) . . . x n with which y is connected 

 by the relation 



there is frequently an option as to the order of magnitude which 

 shall be assigned to the rr's before setting up apparatus. The 

 problem of best magnitudes has long been recognized as a funda- 

 mental one in the theory of the precision of measurements, but 

 available discussions are limited in their usefulness by the assump- 

 tion of constant errors or proceed by a step-by-step calculation 

 invented anew for each problem. 1 



The present analysis aims to formulate the reasoning once for 

 all, setting up a general criterion from which special cases may be 

 derived at will. 



This is accomplished by the consideration of two functions 4> 

 and 6 beside the function / characterizing the instrument or 

 method. If in general Ax denote an error in the observed quan- 



1 See, for example, Holman, "Precision of Measurements," pp. 100-119; 

 Mellor, "Higher Mathematics for Students of Chemistry and Physics," pp. 531 • 

 540; Baum, "General Method for Determining Best Reading of an Instrument 

 for Least Error," Phys. Rev. 1899: pp. 181-182. 



187 



