Empirical Proof of the Law of Error, 335 



Modulus) of the Probability- curve : — 

 c=*l'772...e. 



In testing the fit of these curves we may pursue two me- 

 thods. We may compare computed with experienced values, 

 either with respect to small parts or some large aggregate. 

 For instance, and in particular, we may examine either ordi- 

 nates or areas ; either small strips, or a large extent, of area. 

 The former test yields the greatest quantity of evidence if we 

 take the trouble of extracting it ; but for the same labour the 

 comparison of areas yields a better result than that of ordinates. 



Confining ourselves here to the more summary tests, we 

 may observe that it is not indifferent what point we select in 

 order to observe whether the ordinate given by experience at 

 that point, or the area within it, is better represented by the 

 Probability-, or some rival, curve. Consider in the accom- 

 panying figure the relation of the Probability-curve (the con- 

 tinuous line) to the parabola (the discontinuous line), each 

 being supposed to have the same mean error, in the sense 

 above explained. If we compare real with computed ordinates 

 in the neighbourhood of either the point ^ or i 2 , it is clear 



*-i 



trvz 



1*2, 



that there will not be much to choose between the parabola 

 and the Probability-curve. It would be better to select as the 

 theatres of our comparison the neighbourhood of in x or m 2 , 

 where the difference between the computed ordinates of the 

 rival curves is a maximum. Conversely, i x and i 2 would be 

 good points to select for the comparison of areas. 



It will be desirable to bring to the empirical investigation 

 a knowledge of these critical points, as they may be called, 

 for all the curves with which we are concerned. They may 

 be expressed conveniently in terms of the computed Modulus 

 of the Probability-curve, that is *Jtt6. In the accompanying 



