A NEW GENERAL THEORY OF ERRORS. 

 By William Edward Story. 



Presented April 13, 1904. Received May 5, 1904. 



I. Introduction. 



The ordinary theory of errors — the method of least squares — has 

 been developed in three ways (methods of Gauss, Hagen, and Crofton) ; 

 not only is each of tliese developments based upon assumptions to which 

 not every one will assent, but at certain stages in each of the two later 

 developments exactness is replaced by approximation through the neglect 

 of certain quantities (or terms) regarded as very small (or productive of 

 very little effect) in comparison with others retained, whereas this neglect 

 is not fully justified, as it seems to us. 



The present paper is the outcome of an attempt to construct a theory 

 of errors upon such simple principles as will be generally admitted to be 

 necessar}- for the mathematical treatment of any such theory. The im- 

 mediate object of the theory of errors is to determine the probability that 

 the error made in the direct measurement of a physical magnitude shall 

 lie between given limits. More complex applications of the theory 

 (e. g. to functions of quantities directly observed and to quantities deter- 

 mined by calculations based upon direct observations) can be connected 

 with the solution of the fundamental problem just stated by means 

 familiar to all who have used the ordinary theory. Of course we con- 

 sider only accidental errors, assuming the observations to have been 

 corrected for all systematic errors before being subjected to our investi- 

 gation ; we suppose also that the observations were all made under the 

 same conditions, so that they are equally reliable (their relative prob- 

 abilities must be determined by the theory). 



II. Case op a Finite Number of Observations. 



Let Zi , 2,) 2^3» • • • ? 2!„ be n observed values of a magnitude whose true 

 value is Z, and let z denote an observed value in general. If A is any 

 function of z and A^, Ao, A3, . . . , A,^ are the values of this function for 

 Ci , r, , C3, . . . , 2„, respectively, we write 



