332 Mr. F. Y. Edgeworth on the 



there is secular increase of any quantity, e. g., at some periods, 

 of Bank-notes-in-the-hands-of-the-public, then, the facility- 

 curves not being constant, we shall not look for the Law of 

 Error. Nor, when we have reason to suspect that the facility- 

 curves of the variables are not symmetrical, that there is a 

 drift of tendency in one direction, shall we expect the result- 

 ing arrangement to be symmetrical ; though it is quite possible 

 that the central portion of the generated curve, the body as 

 distinguished from the extremities, may be of the typical 

 form. In no case can the extremities, say the regions at a 

 distance of 1*5 x modulus from the centre, be expected to 

 fulfil the law accurately*. 



The empirical investigation thus directed may conveniently 

 be separated into two steps. We should first ascertain whether, 

 and how far the curve (or class of curves) under consideration 

 is symmetrical ; next, whether, as to that part of it which is 

 symmetrical, it is to be regarded as the Probability- curve and 

 not another. 



I. Whether an observed group is s}^mmetrical or not is in 

 the main to be determined by inspection and common sense. 

 However, the Theory of Probabilities may supply some hints 

 and warnings useful in cases which are not self-evident. One 

 caution is that the position of an extreme observation is not 

 per se a safe test that the limb whose extremity is furthest 

 extended is really longest. Though of course if, examining 

 repeated specimens of a certain category, we find that one 

 limb seems ever longer than the other, this persistent disparity 

 cannot be due to accident. It is thus that the present writer 

 has proved the fact that prices are apt to diverge from their 

 average more in excess than in defectf . Even in that case, 

 however, it was well to take account of the penultimate and 

 antepenultimate, as well as the extreme, observations. 



It is a very good plan, as Dr. Venn has recommended f , to 

 compare the mean-error above and that below the apparent, the 

 arithmetic, mean. The significance of this result may be tested 

 by the formula which Laplace has given for the error of the sum 



tables, that even so small a number of elements will afford a respectable 

 approximation to tbe perfect probability-curve. (" Mathematical Theory 

 of Banking," published in abstract in the Report of the British Associa- 

 tion, 1886.) 



* See illustrations of the last two propositions in my paper " On the 

 Law of Error " &c, Phil. Mag. April 1886. 



i Report of the British Association, 1887 : Memorandum on the best 

 method of measuring the change in the value of money. 



t In an important letter to ' Nature/ Sept. 1, 1887. 



