172 FROCEEDINGS OF THE AMERICAN ACADEMY. 



IV. Effect of an Infinitesimal Source of Error. 



To determine the form of the function ij/ (^) we consider the effect of 

 the addition of an infinitesimal source of error to the original source 

 (liovvever complex that may be), that is, of a source of error that pro- 

 duces infinitesimal errors with great frequency and finite errors of even 

 moderate magnitude with infiiiitesiujal frequency. We employ the same 

 notation for all functions and quantities pertaining to tlie action of the 

 infinitesimal source alone as for those pertaining to the original source 

 alone, distinguishing them by an accent, and denote changes due to the 

 addition of the infinitesimal source by the symbols for the functions and 

 quantities changed with a 8 prefixed. 



A oiven error x produced by the simultaneous action of both sources 

 is the result of a certain error x' produced by the infinitesimal source and 

 an error x — x' produced by the original source. The probability that 

 the error produced by the combination shall lie between x and x -\- dx 

 is, then, 



5-' 



[c^ {x) + 8</> (x)] dx= j (fi{x-x') ■ <{>' (x') • dx ■ dx', 

 from which follows 

 (18) <fi (x) + S4i{x) = I 4>(x- x>) • c^' (x') • dx'. 



x' 



For the arithmetical mean of the error produced by the combination 

 we have 



'x-\-ix x' 



XQ-\-hxQ= I X • dx • I (f>{x — x') • <f)' (x') ■ dx' 



x-\-Sx x' 



X x^ 



= Cx-dx- f4>(x-x')-c}>'(x')-dx'-^x-<f>(x)'Sx-x-cl,(xJ'8x. 

 .c x' 



It is to be observed that, in whatever manner these integrals are 

 evaluated, x — x' must be taken over the whole range of values from x 

 to X. The limits of possible error (x and x) cannot be fixed a priori ; 

 some may think that large errors or residuals are impossible by the nature 

 of accurate observation, but we prefer to regard the limits as determined 

 by the frequency function <^ (x), namely, we assume that the value of 



