170 PROCEEDINGS OF THE AMERICAN ACADEMY. 



III. Case of an Infinite Number of Observations — 

 Assumptions. 



The assumptions upon which we shall base the general theory of errors 

 (upon which the ordinary theory has always been based in part, explicitly 

 or implicitly) are these : 



I. Possible errors form a practically continuous sequence from a certain 

 loioer limit to a certain upper limit ; 



II. The probability that the error of an observation lies between x and 

 X + dx, where dx is infinitesimal, is <^(x) • dx, where 4>(x) is an analyt- 

 ical function of x developable by Taylor s theorem throughout the u'/iole 

 range of possible error ; 



III. The probability that the error lies between given limits is independ- 

 ent of the unit of measurement. 



Assumption I forms the foundation for II, and II implies that 

 (f) (x + h) is developable according to powers of h by Taylor's theorem 

 for all values of x and x -\- h that lie within the range of possible error. 

 The value of ^ (a:) for any given value of x is called the relative fre- 

 quency of the error x. 



The probability that the error of an observation lies between two given 

 limits a (lower) and b (upper) is, evidently, 



6 



(8) / <^ (a^) • ^^- 



a 



If X is the (algebraically) least possible error and x the greatest possible 

 error, we have 



(9) fi]> (x) ■ dx=l. 



X 



If we express x in terms of the residual ^ by (3), representing the lower 

 and upper limits of possible residuals by | and |, respectively (which 

 correspond to the extreme errors x and x), and writing 



(10) c^ (x) = </, (^ + X,) = ^{e), 



we have from (9), because Xq is constant, 

 (11) JV(0-rf^=l, 



