462 Averages. [CHAP. xviu. 



determination of the mean error would give but a slight 

 intimation of the sort of outline of our Curve of Facility. 

 We might then have found it convenient to adopt some plan 

 of successive approximation, by adding a third or fourth 

 mean. Just as we assign the mean value of the magni 

 tude, and its mean departure from this mean ; so we might 

 take this mean error (however determined) as a fresh starting 

 point, and assign the mean departure from it. If the point 

 were worth further discussion we might easily illustrate by 

 means of a diagram the sort of successive approximations 

 which such indications would yield as to the ultimate form 

 of the Curve of Facility or Law of Error. 



As this volume is written mainly for those who take an interest in the 

 logical questions involved, rather than as an introduction to the actual 

 processes of calculation, mathematical details have been throughout avoided 

 as much as possible. For this reason comparatively few references have 

 been made to the exponential equation of the Law of Error, or to the 

 corresponding Probability integral, tables of which are given in several 

 handbooks on the subject. There are two points however in connection 

 with these particular topics as to which difficulties are, or should be, felt by 

 so many students that some notice may be taken of them here 



(1) In regard to the ordinary algebraical expression for the law of error, 



viz. y = p e~ h2x2 , it will have been observed that I have always spoken of y 

 X/TT 



as being proportional to the number of errors of the particular magnitude x. 

 It would hardly be correct to say, absolutely, that y represents that number, 

 because of course the actual number of errors of any precise magnitude, 

 where continuity of possibility is assumed, must be indefinitely small. If 

 therefore we want to pass from the continuous to the discrete, by ascertaining 

 the actual number of errors between two consecutive divisions of our scale, 

 when, as usual in measurements, all within certain limits are referred to 

 some one precise point, we must modify our formula. In accordance with 

 the usual differential notation, we must say that the number of errors falling 



into one subdivision (dx) of our scale is dx e~ h2x2 , where dx is a (small) 



X/TT 

 unit of length, in which both Ir 1 and x must be measured. 



