for the Theory of Errors. 489 



in saying in which direction the error lies, and therefore no 

 motive in altering the adjustment. Now the bias and want 

 of acuteness of our judgment will of course be variable, the 

 circumstances which affect them (experience, fatigue, illumi- 

 nation, &c.) being almost endless ; but I think this principle 

 may be conceded : If an index he placed in a given position 

 and examined from a given point of view, we shall judge the 

 adjustment to be correct when, but not unless, the error falls 

 between certain limits ; such limits being determined by sub- 

 jective and other conditions at the time of the observation. 



12. Now consider (merely for the sake of illustration) 

 two ways in which an adjustment may be made. Suppose 

 we are measuring the positions of spectrum-lines by bringing 

 a wire into coincidence with them, and disregarding paral- 

 lactic and instrumental errors, let us confine our attention to 

 errors of judgment in adjusting the cross-wire. First let the 

 determinations be made in this (impracticable) manner : the 

 micrometer head is first moved at random, and then the field 

 observed, and if, in the judgment of the observer, the vertical 

 wire coincides accurately with a spectrum-line, the micrometer 

 reading is recorded. Whenever an apparent coincidence is 

 registered, the error of judgment must have fallen between 

 certain limits ; and obviously it is equally likely to have 

 any value between those limits. In a number of observa- 

 tions the law of error may be determined as follows : — Draw 

 a number of horizontal lines (fig. 13) each representing on a 

 convenient scale the limits of possi- 

 ble error at the time of an observa- 

 tion*. The relative frequency of 

 an error N will be proportional 

 to the number of horizontal lines cut 

 by the ordinate P N. In general 

 the curve of error for a large num- 

 ber of observations will be straight and horizontal in the 

 middle, falling down towards the ends so as to meet the 

 axis of errors ; if the limits of errors of judgment were 

 invariable, the curve of error would be of the type of fig. 1. 



13. Perhaps in practice we shall more nearly approximate 

 to the following plan : — Starting with the wire in a position 

 which is evidently in error, we shall move it up to the 

 spectrum- line until no further error can be detected. If we 

 really stop moving the wire as soon as we cease to feel a 

 distinct motive for doing so, we shall be just at the limit of 

 possible error corresponding to that time. For a series of 



* What we really require to know is the law of frequency for the limits 

 of error. 



Fig. 



13. 



J 



3 































r 



si 



